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Derivada de:
Derivada de 1/(x+3)
Derivada de x^(5*x)
Derivada de x^3/6
Derivada de x^-8
Expresiones idénticas
y=(e^(-15x*sinx^ tres))/((cos^ tres)lg*x^ dos)
y es igual a (e en el grado ( menos 15x multiplicar por seno de x al cubo )) dividir por (( coseno de al cubo )lg multiplicar por x al cuadrado )
y es igual a (e en el grado ( menos 15x multiplicar por seno de x en el grado tres)) dividir por (( coseno de en el grado tres)lg multiplicar por x en el grado dos)
y=(e(-15x*sinx3))/((cos3)lg*x2)
y=e-15x*sinx3/cos3lg*x2
y=(e^(-15x*sinx³))/((cos³)lg*x²)
y=(e en el grado (-15x*sinx en el grado 3))/((cos en el grado 3)lg*x en el grado 2)
y=(e^(-15xsinx^3))/((cos^3)lgx^2)
y=(e(-15xsinx3))/((cos3)lgx2)
y=e-15xsinx3/cos3lgx2
y=e^-15xsinx^3/cos^3lgx^2
y=(e^(-15x*sinx^3)) dividir por ((cos^3)lg*x^2)
Expresiones semejantes
y=(e^(15x*sinx^3))/((cos^3)lg*x^2)
Expresiones con funciones
Coseno cos
cos(x^(2/3))/(sin(3*x)+2^x)
cos(sqrt(4*x/log(x)))
cos(√sin(tanπx))
cos√(sin(tan(πx)))
cos(b)cot(b)

Derivada de la función/ y=(e^(-15x*sinx^3))/((cos^3)lg*x^2)

Derivada de y=(e^(-15xsinx^3))/((cos^3)lgx^2)

Solución

Ha introducido [src]

           3   
  -15*x*sin (x)
 E             
---------------
   3       2   
cos (x)*log (x)

$$\frac{e^{- 15 x \sin^{3}{\left(x \right)}}}{\log{\left(x \right)}^{2} \cos^{3}{\left(x \right)}}$$

E^((-15*x)*sin(x)^3)/((cos(x)^3*log(x)^2))

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = e^{- 15 x \sin^{3}{\left(x \right)}}$ y $g{\left(x \right)} = \log{\left(x \right)}^{2} \cos^{3}{\left(x \right)}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Sustituimos $u = - 15 x \sin^{3}{\left(x \right)}$ .
2. Derivado $e^{u}$ es.
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} - 15 x \sin^{3}{\left(x \right)}$ :
  1. Se aplica la regla de la derivada de una multiplicación:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = - 15 x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
      1. Según el principio, aplicamos: $x$ tenemos $1$
      Entonces, como resultado: $-15$
    $g{\left(x \right)} = \sin^{3}{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. Sustituimos $u = \sin{\left(x \right)}$ .
    2. Según el principio, aplicamos: $u^{3}$ tenemos $3 u^{2}$
    3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(x \right)}$ :
      1. La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      Como resultado de la secuencia de reglas:
      
      $3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
    Como resultado de: $- 45 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 15 \sin^{3}{\left(x \right)}$
  Como resultado de la secuencia de reglas:
  
  $\left(- 45 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 15 \sin^{3}{\left(x \right)}\right) e^{- 15 x \sin^{3}{\left(x \right)}}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \cos^{3}{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Sustituimos $u = \cos{\left(x \right)}$ .
  2. Según el principio, aplicamos: $u^{3}$ tenemos $3 u^{2}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Como resultado de la secuencia de reglas:
    
    $- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
  $g{\left(x \right)} = \log{\left(x \right)}^{2}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = \log{\left(x \right)}$ .
  2. Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \log{\left(x \right)}$ :
    1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
    Como resultado de la secuencia de reglas:
    
    $\frac{2 \log{\left(x \right)}}{x}$
  Como resultado de: $- 3 \log{\left(x \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{2 \log{\left(x \right)} \cos^{3}{\left(x \right)}}{x}$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{\left(- 45 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 15 \sin^{3}{\left(x \right)}\right) e^{- 15 x \sin^{3}{\left(x \right)}} \log{\left(x \right)}^{2} \cos^{3}{\left(x \right)} - \left(- 3 \log{\left(x \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{2 \log{\left(x \right)} \cos^{3}{\left(x \right)}}{x}\right) e^{- 15 x \sin^{3}{\left(x \right)}}}{\log{\left(x \right)}^{4} \cos^{6}{\left(x \right)}}$
Simplificamos:

$\frac{\left(- 15 x \left(3 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 3 x \log{\left(x \right)} \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) e^{- 15 x \sin^{3}{\left(x \right)}}}{x \log{\left(x \right)}^{3} \cos^{4}{\left(x \right)}}$

Respuesta:

$\frac{\left(- 15 x \left(3 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 3 x \log{\left(x \right)} \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) e^{- 15 x \sin^{3}{\left(x \right)}}}{x \log{\left(x \right)}^{3} \cos^{4}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

                                                                      /       3                                     \           3   
                                                                      |  2*cos (x)*log(x)        2       2          |  -15*x*sin (x)
                                                               3      |- ---------------- + 3*cos (x)*log (x)*sin(x)|*e             
       1        /        3              2          \  -15*x*sin (x)   \         x                                   /               
---------------*\- 15*sin (x) - 45*x*sin (x)*cos(x)/*e              + --------------------------------------------------------------
   3       2                                                                                    6       4                           
cos (x)*log (x)                                                                              cos (x)*log (x)

$$\frac{1}{\log{\left(x \right)}^{2} \cos^{3}{\left(x \right)}} \left(- 45 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 15 \sin^{3}{\left(x \right)}\right) e^{- 15 x \sin^{3}{\left(x \right)}} + \frac{\left(3 \log{\left(x \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} - \frac{2 \log{\left(x \right)} \cos^{3}{\left(x \right)}}{x}\right) e^{- 15 x \sin^{3}{\left(x \right)}}}{\log{\left(x \right)}^{4} \cos^{6}{\left(x \right)}}$$

Simplificar

Segunda derivada [src]

/                                                                                                                                                                               2                               2                                                                                                                                                                                   \               
|                                                                                                                                                           2       2      2*cos (x)        2       2      2*cos (x)*log(x)   12*cos(x)*log(x)*sin(x)                                                                                                                                               |               
|                                                                                                                                                    - 6*log (x)*sin (x) - --------- + 3*cos (x)*log (x) + ---------------- + -----------------------     /  2*cos(x)                  \     /  2*cos(x)                  \                                                                         |               
|                                                                                                                                                                               2                                  2                     x              2*|- -------- + 3*log(x)*sin(x)|   3*|- -------- + 3*log(x)*sin(x)|*sin(x)                                                                  |               
|                                                                                           /     2       3*sin(x)\ /  2*cos(x)                  \                             x                                  x                                       \     x                      /     \     x                      /                2    /  2*cos(x)                  \                      |               
|                                                                                           |- -------- + --------|*|- -------- + 3*log(x)*sin(x)| + ------------------------------------------------------------------------------------------------ - -------------------------------- + ---------------------------------------   30*sin (x)*|- -------- + 3*log(x)*sin(x)|*(3*x*cos(x) + sin(x))|           3   
|   /     2             2                                               2    3   \          \  x*log(x)    cos(x) / \     x                      /                                            cos(x)*log(x)                                                         x*log(x)                                cos(x)                              \     x                      /                      |  -15*x*sin (x)
|45*\x*sin (x) - 2*x*cos (x) - 2*cos(x)*sin(x) + 5*(3*x*cos(x) + sin(x)) *sin (x)/*sin(x) + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------|*e             
\                                                                                                                                                                                                       cos(x)*log(x)                                                                                                                                         cos(x)*log(x)                         /               
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                     3       2                                                                                                                                                                                                      
                                                                                                                                                                                                  cos (x)*log (x)

$$\frac{\left(- \frac{30 \left(3 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \sin^{2}{\left(x \right)}}{\log{\left(x \right)} \cos{\left(x \right)}} + 45 \left(x \sin^{2}{\left(x \right)} - 2 x \cos^{2}{\left(x \right)} + 5 \left(3 x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \sin^{3}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin{\left(x \right)} + \frac{\left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \left(\frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{2}{x \log{\left(x \right)}}\right) + \frac{3 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{- 6 \log{\left(x \right)}^{2} \sin^{2}{\left(x \right)} + 3 \log{\left(x \right)}^{2} \cos^{2}{\left(x \right)} + \frac{12 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{2 \log{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} - \frac{2 \cos^{2}{\left(x \right)}}{x^{2}}}{\log{\left(x \right)} \cos{\left(x \right)}} - \frac{2 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right)}{x \log{\left(x \right)}}}{\log{\left(x \right)} \cos{\left(x \right)}}\right) e^{- 15 x \sin^{3}{\left(x \right)}}}{\log{\left(x \right)}^{2} \cos^{3}{\left(x \right)}}$$

Simplificar

Tercera derivada [src]

/                                                                                                                                                                                                                       3                                                           3                   3                   2                   2                          2                                                                                                                                                               /                           2                               2                                    \     /                           2                               2                                    \                                                                                                                                  /                           2                               2                                    \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   \               
|                                                                                                                                                                                                                  6*cos (x)        2       3            2       2             4*cos (x)*log(x)   18*cos (x)*log(x)   18*cos (x)*sin(x)   36*sin (x)*cos(x)*log(x)   18*cos (x)*log(x)*sin(x)                                                                                                                      /     2       3*sin(x)\ |       2       2      2*cos (x)        2       2      2*cos (x)*log(x)   12*cos(x)*log(x)*sin(x)|     |       2       2      2*cos (x)        2       2      2*cos (x)*log(x)   12*cos(x)*log(x)*sin(x)|                                                                                                                                  |       2       2      2*cos (x)        2       2      2*cos (x)*log(x)   12*cos(x)*log(x)*sin(x)|                                                                                                                                                                                                /                                                                                    2                               2                                                                                                                 \                                                                                                                           |               
|                                                                                                                                                                                                                  --------- + 6*log (x)*sin (x) - 21*cos (x)*log (x)*sin(x) - ---------------- + ----------------- + ----------------- - ------------------------ - ------------------------     /  2*cos(x)                  \      /  2*cos(x)                  \         2    /  2*cos(x)                  \   |- -------- + --------|*|- 6*log (x)*sin (x) - --------- + 3*cos (x)*log (x) + ---------------- + -----------------------|   6*|- 6*log (x)*sin (x) - --------- + 3*cos (x)*log (x) + ---------------- + -----------------------|     /     2       3*sin(x)\ /  2*cos(x)                  \     /     2       3*sin(x)\ /  2*cos(x)                  \          9*|- 6*log (x)*sin (x) - --------- + 3*cos (x)*log (x) + ---------------- + -----------------------|*sin(x)      /  2*cos(x)                  \                                                                                                                                                     |                                                                2       2      2*cos (x)        2       2      2*cos (x)*log(x)   12*cos(x)*log(x)*sin(x)                                                                             |                                                                                                                           |               
|                                                                                                                       /                                   2                     \                                     3                                                              3                  x                    2                     x                           2              2*|- -------- + 3*log(x)*sin(x)|   10*|- -------- + 3*log(x)*sin(x)|   21*sin (x)*|- -------- + 3*log(x)*sin(x)|   \  x*log(x)    cos(x) / |                           2                                  2                     x           |     |                           2                                  2                     x           |   2*|- -------- + --------|*|- -------- + 3*log(x)*sin(x)|   3*|- -------- + --------|*|- -------- + 3*log(x)*sin(x)|*sin(x)     |                           2                                  2                     x           |          24*|- -------- + 3*log(x)*sin(x)|*sin(x)                                                                                                                                              |                                                         - 6*log (x)*sin (x) - --------- + 3*cos (x)*log (x) + ---------------- + -----------------------     /  2*cos(x)                  \     /  2*cos(x)                  \       |                                                                                                                           |               
|                                                                                        /  2*cos(x)                  \ |        2           6        12*sin (x)      12*sin(x)   |   6*cos(x)                         x                                                              x                                       x                                                 x                 \     x                      /      \     x                      /              \     x                      /                           \                          x                                  x                                  /     \                          x                                  x                                  /     \  x*log(x)    cos(x) / \     x                      /     \  x*log(x)    cos(x) / \     x                      /            \                          x                                  x                                  /             \     x                      /                                                                                                                                                     |                                                                                    2                                  2                     x              2*|- -------- + 3*log(x)*sin(x)|   3*|- -------- + 3*log(x)*sin(x)|*sin(x)|                                                                                                                           |               
|                                                                                        |- -------- + 3*log(x)*sin(x)|*|3 + --------- + ---------- + ---------- - ---------------| - -------- + 9*log(x)*sin(x) + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------- + --------------------------------- + ----------------------------------------- + -------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------- - -------------------------------------------------------- + --------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------- - ----------------------------------------                                                                                                                   2                          |/     2       3*sin(x)\ /  2*cos(x)                  \                             x                                  x                                       \     x                      /     \     x                      /       |       /  2*cos(x)                  \ /     2             2                                               2    3   \       |               
|                                                                                        \     x                      / |     2           2    2          2        x*cos(x)*log(x)|      x                                                                                                          2                                                                                                       2                                   2    2                                     2                                                                             cos(x)*log(x)                                                                                                                2                                                                        x*log(x)                                                        cos(x)                                                                                 2                                                                       x*cos(x)*log(x)                                                                                                                          45*sin (x)*(3*x*cos(x) + sin(x))*||- -------- + --------|*|- -------- + 3*log(x)*sin(x)| + ------------------------------------------------------------------------------------------------ - -------------------------------- + ---------------------------------------|   135*|- -------- + 3*log(x)*sin(x)|*\x*sin (x) - 2*x*cos (x) - 2*cos(x)*sin(x) + 5*(3*x*cos(x) + sin(x)) *sin (x)/*sin(x)|           3   
|       3                                3    6             2                     3                                     \    x *log(x)   x *log (x)    cos (x)                    /                                                                                                              cos (x)*log(x)                                                                                            x *log(x)                           x *log (x)                               cos (x)                                                                                                                                                                                           x*cos(x)*log (x)                                                                                                                                                                                                                         cos (x)*log(x)                                                                                                     2                     3                          /       2             2                     \                                    \\  x*log(x)    cos(x) / \     x                      /                                            cos(x)*log(x)                                                         x*log(x)                                cos(x)                /       \     x                      /                                                                                      |  -15*x*sin (x)
|135*sin (x) - 3375*(3*x*cos(x) + sin(x)) *sin (x) - 270*cos (x)*sin(x) - 90*x*cos (x) + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + 315*x*sin (x)*cos(x) + 2025*sin (x)*(3*x*cos(x) + sin(x))*\- x*sin (x) + 2*x*cos (x) + 2*cos(x)*sin(x)/ - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------|*e             
\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          cos(x)*log(x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             cos(x)*log(x)                                                                                                                                                                                      cos(x)*log(x)                                                      /               
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 3       2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              cos (x)*log (x)

$$\frac{\left(315 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 90 x \cos^{3}{\left(x \right)} - 3375 \left(3 x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \sin^{6}{\left(x \right)} + 2025 \left(3 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(- x \sin^{2}{\left(x \right)} + 2 x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin^{3}{\left(x \right)} - \frac{45 \left(3 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \left(\frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{2}{x \log{\left(x \right)}}\right) + \frac{3 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{- 6 \log{\left(x \right)}^{2} \sin^{2}{\left(x \right)} + 3 \log{\left(x \right)}^{2} \cos^{2}{\left(x \right)} + \frac{12 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{2 \log{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} - \frac{2 \cos^{2}{\left(x \right)}}{x^{2}}}{\log{\left(x \right)} \cos{\left(x \right)}} - \frac{2 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right)}{x \log{\left(x \right)}}\right) \sin^{2}{\left(x \right)}}{\log{\left(x \right)} \cos{\left(x \right)}} + \frac{135 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \left(x \sin^{2}{\left(x \right)} - 2 x \cos^{2}{\left(x \right)} + 5 \left(3 x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \sin^{3}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\log{\left(x \right)} \cos{\left(x \right)}} + \frac{\frac{3 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \left(\frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{2}{x \log{\left(x \right)}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \left(\frac{12 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 3 - \frac{12 \sin{\left(x \right)}}{x \log{\left(x \right)} \cos{\left(x \right)}} + \frac{2}{x^{2} \log{\left(x \right)}} + \frac{6}{x^{2} \log{\left(x \right)}^{2}}\right) + \frac{21 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\left(\frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{2}{x \log{\left(x \right)}}\right) \left(- 6 \log{\left(x \right)}^{2} \sin^{2}{\left(x \right)} + 3 \log{\left(x \right)}^{2} \cos^{2}{\left(x \right)} + \frac{12 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{2 \log{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} - \frac{2 \cos^{2}{\left(x \right)}}{x^{2}}\right)}{\log{\left(x \right)} \cos{\left(x \right)}} + \frac{9 \left(- 6 \log{\left(x \right)}^{2} \sin^{2}{\left(x \right)} + 3 \log{\left(x \right)}^{2} \cos^{2}{\left(x \right)} + \frac{12 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{2 \log{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} - \frac{2 \cos^{2}{\left(x \right)}}{x^{2}}\right) \sin{\left(x \right)}}{\log{\left(x \right)} \cos^{2}{\left(x \right)}} + \frac{6 \log{\left(x \right)}^{2} \sin^{3}{\left(x \right)} - 21 \log{\left(x \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} - \frac{36 \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{18 \log{\left(x \right)} \cos^{3}{\left(x \right)}}{x} - \frac{18 \log{\left(x \right)} \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} + \frac{18 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} - \frac{4 \log{\left(x \right)} \cos^{3}{\left(x \right)}}{x^{3}} + \frac{6 \cos^{3}{\left(x \right)}}{x^{3}}}{\log{\left(x \right)} \cos^{2}{\left(x \right)}} + 9 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \left(\frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{2}{x \log{\left(x \right)}}\right)}{x \log{\left(x \right)}} - \frac{24 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{x \log{\left(x \right)} \cos{\left(x \right)}} - \frac{6 \left(- 6 \log{\left(x \right)}^{2} \sin^{2}{\left(x \right)} + 3 \log{\left(x \right)}^{2} \cos^{2}{\left(x \right)} + \frac{12 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{2 \log{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} - \frac{2 \cos^{2}{\left(x \right)}}{x^{2}}\right)}{x \log{\left(x \right)}^{2} \cos{\left(x \right)}} - \frac{6 \cos{\left(x \right)}}{x} + \frac{2 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right)}{x^{2} \log{\left(x \right)}} + \frac{10 \left(3 \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right)}{x^{2} \log{\left(x \right)}^{2}}}{\log{\left(x \right)} \cos{\left(x \right)}} + 135 \sin^{3}{\left(x \right)} - 270 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right) e^{- 15 x \sin^{3}{\left(x \right)}}}{\log{\left(x \right)}^{2} \cos^{3}{\left(x \right)}}$$

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Gráfico

Derivada de y=(e^(-15x*sinx^3))/((cos^3)lg*x^2)

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Derivada de y=(e^(-15xsinx^3))/((cos^3)lgx^2)

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Derivada de y=(e^(-15x*sinx^3))/((cos^3)lg*x^2)

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Definida a trozos:

Solución

Derivada de y=(e^(-15xsinx^3))/((cos^3)lgx^2)