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Derivada de -5*tan(2*t)-4*cot(4*t)
Derivada de (3*sqrt(v)-2*v*e^v)/v
Derivada de 3^(x*x)
Derivada de 2*x-8/x
Expresiones idénticas
y=ln(sinxcosx/(e^x) uno /(dos ^x))
y es igual a ln( seno de x coseno de x dividir por (e en el grado x)1 dividir por (2 en el grado x))
y es igual a ln( seno de x coseno de x dividir por (e en el grado x) uno dividir por (dos en el grado x))
y=ln(sinxcosx/(ex)1/(2x))
y=lnsinxcosx/ex1/2x
y=lnsinxcosx/e^x1/2^x
y=ln(sinxcosx dividir por (e^x)1 dividir por (2^x))

Derivada de la función/ y=ln(sinxcosx/(e^x)1/(2^x))

Derivada de y=ln(sinxcosx/(e^x)1/(2^x))

Solución

Ha introducido [src]

   //sin(x)*cos(x)\\
   ||-------------||
   ||       x     ||
   |\      E      /|
log|---------------|
   |        x      |
   \       2       /

$$\log{\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)} \frac{1}{e^{x}}}{2^{x}} \right)}$$

log(((sin(x)*cos(x))/E^x)/2^x)

Solución detallada

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

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

$- 2 \tan{\left(x \right)} - 1 - \log{\left(2 \right)} + \frac{2}{\sin{\left(2 x \right)}}$

Respuesta:

$- 2 \tan{\left(x \right)} - 1 - \log{\left(2 \right)} + \frac{2}{\sin{\left(2 x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

 x / -x //   2         2   \  -x           -x       \    -x         -x              \  x
2 *\2  *\\cos (x) - sin (x)/*e   - cos(x)*e  *sin(x)/ - 2  *cos(x)*e  *log(2)*sin(x)/*e 
----------------------------------------------------------------------------------------
                                     cos(x)*sin(x)

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

Simplificar

Segunda derivada [src]

   2         2                                                2         2                                                    2           2         2             /   2         2                   \             2                                  2                                              2         2                                             /   2         2                                          \       
sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)   sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)   - 2*cos (x) + 2*sin (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) - 3*cos(x)*sin(x) + log (2)*cos(x)*sin(x) + cos(x)*log(2)*sin(x)   sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)   \sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)/*log(2)
-------------------------------------------------------- - -------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------- - -----------------------------------------------------------------
                           2                                                          2                                                                                                            cos(x)*sin(x)                                                                                                     cos(x)*sin(x)                                                   cos(x)*sin(x)                          
                        sin (x)                                                    cos (x)

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

Simplificar

Tercera derivada [src]

    /   2         2                                          \     /       2           2         2             /   2         2                   \             2                                  2                                        \     /       2           2         2             /   2         2                   \             2                                  2                                        \     /   2         2                                          \      2         2                                                2         2         2    /   2         2                   \                           2       2           2               /       2           2                     \               2       2           2                3                                                  2                      /   2         2                                          \            /   2         2                                          \            /   2         2                                          \            /       2           2         2             /   2         2                   \             2                                  2                                        \     /   2         2                                          \             2    /   2         2                                          \     /   2         2                                          \            /       2           2         2             /   2         2                   \             2                                  2                                        \       
  2*\sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)/   2*\- 2*cos (x) + 2*sin (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) - 3*cos(x)*sin(x) + log (2)*cos(x)*sin(x) + cos(x)*log(2)*sin(x)/   2*\- 2*cos (x) + 2*sin (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) - 3*cos(x)*sin(x) + log (2)*cos(x)*sin(x) + cos(x)*log(2)*sin(x)/   2*\sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)/   sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)   cos (x) - sin (x) + log (2)*\sin (x) - cos (x) + cos(x)*sin(x)/ - 11*cos(x)*sin(x) - 2*cos (x)*log (2) - 2*cos (x)*log(2) - 2*\- 2*sin (x) + 2*cos (x) + 3*cos(x)*sin(x)/*log(2) + 2*log (2)*sin (x) + 2*sin (x)*log(2) + log (2)*cos(x)*sin(x) - 3*cos(x)*log(2)*sin(x) + 2*log (2)*cos(x)*sin(x)   2*\sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)/*sin(x)   2*\sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)/*log(2)   2*\sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)/*cos(x)   2*\- 2*cos (x) + 2*sin (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) - 3*cos(x)*sin(x) + log (2)*cos(x)*sin(x) + cos(x)*log(2)*sin(x)/   2*\sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)/*log(2)   log (2)*\sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)/   2*\sin (x) - cos (x) + cos(x)*sin(x) + cos(x)*log(2)*sin(x)/*log(2)   2*\- 2*cos (x) + 2*sin (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) - 3*cos(x)*sin(x) + log (2)*cos(x)*sin(x) + cos(x)*log(2)*sin(x)/*log(2)
- ------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------ - -------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------- - ------------------------------------------------------------------- - ------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------- - ------------------------------------------------------------------ - ------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                               2                                                                                                                      2                                                                                                                                                                             2                                                                                                                     2                                                      cos(x)*sin(x)                                                                                                                                                                   cos(x)*sin(x)                                                                                                                                                                               3                                                                     2                                                                     3                                                                                                                   cos(x)*sin(x)                                                                                                                   2                                                              cos(x)*sin(x)                                                         cos(x)*sin(x)                                                                                                                cos(x)*sin(x)                                                                                   
                            cos (x)                                                                                                                sin (x)                                                                                                                                                                       cos (x)                                                                                                               sin (x)                                                                                                                                                                                                                                                                                                                                                                                                                            cos (x)                                                               cos (x)                                                               sin (x)                                                                                                                                                                                                                                             sin (x)

$$- \frac{2 \left(\sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\cos^{2}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}^{2}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} + 2 \sin^{2}{\left(x \right)} - 3 \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} + 2 \sin^{2}{\left(x \right)} - 3 \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} + 2 \sin^{2}{\left(x \right)} - 3 \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} + 2 \sin^{2}{\left(x \right)} - 3 \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)}} - \frac{- 2 \left(- 2 \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} + \left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}^{2} - \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)}^{2} \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)} \cos{\left(x \right)} - 3 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)}^{3} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)} - 2 \log{\left(2 \right)}^{2} \cos^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$

Simplificar

Gráfico

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Derivada de y=ln(sinxcosx/(e^x)1/(2^x))

Gráfico:

Definida a trozos:

Solución