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Derivada de:
Derivada de x^4/3-x
Derivada de x^-4/5
Derivada de x=1
Derivada de x^2*2^x
Expresiones idénticas
y= tres ^sqrt(x)cos^ tres (tgx)
y es igual a 3 en el grado raíz cuadrada de (x) coseno de al cubo (tgx)
y es igual a tres en el grado raíz cuadrada de (x) coseno de en el grado tres (tgx)
y=3^√(x)cos^3(tgx)
y=3sqrt(x)cos3(tgx)
y=3sqrtxcos3tgx
y=3^sqrt(x)cos³(tgx)
y=3 en el grado sqrt(x)cos en el grado 3(tgx)
y=3^sqrtxcos^3tgx
Expresiones con funciones
Coseno cos
cos(3x^2+2)

Derivada de la función/ cos^3/ (tgx)/ sqrt(x)/ y=3^sqrt(x)cos^3(tgx)

Derivada de y=3^sqrt(x)cos^3(tgx)

Solución

Ha introducido [src]

   ___             
 \/ x     3        
3     *cos (tan(x))

$$3^{\sqrt{x}} \cos^{3}{\left(\tan{\left(x \right)} \right)}$$

3^(sqrt(x))*cos(tan(x))^3

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = 3^{\sqrt{x}}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Sustituimos $u = \sqrt{x}$ .
2. $\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sqrt{x}$ :
  1. Según el principio, aplicamos: $\sqrt{x}$ tenemos $\frac{1}{2 \sqrt{x}}$
  Como resultado de la secuencia de reglas:
  
  $\frac{3^{\sqrt{x}} \log{\left(3 \right)}}{2 \sqrt{x}}$
$g{\left(x \right)} = \cos^{3}{\left(\tan{\left(x \right)} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Sustituimos $u = \cos{\left(\tan{\left(x \right)} \right)}$ .
2. Según el principio, aplicamos: $u^{3}$ tenemos $3 u^{2}$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos{\left(\tan{\left(x \right)} \right)}$ :
  1. Sustituimos $u = \tan{\left(x \right)}$ .
  2. La derivada del coseno es igual a menos el seno:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
    1. Reescribimos las funciones para diferenciar:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
    2. Se aplica la regla de la derivada parcial:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
      
      Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
      1. La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
      1. La derivada del coseno es igual a menos el seno:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      Ahora aplicamos la regla de la derivada de una divesión:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    Como resultado de la secuencia de reglas:
    
    $- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}}$
  Como resultado de la secuencia de reglas:
  
  $- \frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan{\left(x \right)} \right)} \cos^{2}{\left(\tan{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}}$
Como resultado de: $- \frac{3 \cdot 3^{\sqrt{x}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan{\left(x \right)} \right)} \cos^{2}{\left(\tan{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \frac{3^{\sqrt{x}} \log{\left(3 \right)} \cos^{3}{\left(\tan{\left(x \right)} \right)}}{2 \sqrt{x}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

   ___                                                                      
 \/ x     3                       ___                                       
3     *cos (tan(x))*log(3)      \/ x     2         /       2   \            
-------------------------- - 3*3     *cos (tan(x))*\1 + tan (x)/*sin(tan(x))
             ___                                                            
         2*\/ x

$$- 3 \cdot 3^{\sqrt{x}} \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan{\left(x \right)} \right)} \cos^{2}{\left(\tan{\left(x \right)} \right)} + \frac{3^{\sqrt{x}} \log{\left(3 \right)} \cos^{3}{\left(\tan{\left(x \right)} \right)}}{2 \sqrt{x}}$$

Simplificar

Segunda derivada [src]

       /                                                                                                                      2         /   1     log(3)\                                                        \            
       |                                                                                                                   cos (tan(x))*|- ---- + ------|*log(3)                                                 |            
   ___ |                                                                                                                                |   3/2     x   |            /       2   \                               |            
 \/ x  |    /       2   \ /   2         /       2   \        2         /       2   \                                   \                \  x            /          3*\1 + tan (x)/*cos(tan(x))*log(3)*sin(tan(x))|            
3     *|- 3*\1 + tan (x)/*\cos (tan(x))*\1 + tan (x)/ - 2*sin (tan(x))*\1 + tan (x)/ + 2*cos(tan(x))*sin(tan(x))*tan(x)/ + ------------------------------------- - ----------------------------------------------|*cos(tan(x))
       |                                                                                                                                     4                                           ___                     |            
       \                                                                                                                                                                               \/ x                      /

$$3^{\sqrt{x}} \left(\frac{\left(\frac{\log{\left(3 \right)}}{x} - \frac{1}{x^{\frac{3}{2}}}\right) \log{\left(3 \right)} \cos^{2}{\left(\tan{\left(x \right)} \right)}}{4} - 3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(\tan{\left(x \right)} \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(\tan{\left(x \right)} \right)} + 2 \sin{\left(\tan{\left(x \right)} \right)} \cos{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(\tan{\left(x \right)} \right)} \cos{\left(\tan{\left(x \right)} \right)}}{\sqrt{x}}\right) \cos{\left(\tan{\left(x \right)} \right)}$$

Simplificar

Tercera derivada [src]

       /                                                                                                                                                                                                                                                                                      /          2              \                                                                                                                                                                                                               \
       |                                                                                                                                                                                                                                                                            3         | 3     log (3)   3*log(3)|                                                                                                                                                   2         /       2   \ /   1     log(3)\                   |
       |                                                                                                                                                                                                                                                                         cos (tan(x))*|---- + ------- - --------|*log(3)                                                                                                                                       9*cos (tan(x))*\1 + tan (x)/*|- ---- + ------|*log(3)*sin(tan(x))|
   ___ |                  /               2                               2                                                                                                                                                                                                  \                | 5/2      3/2        2   |            /       2   \ /   2         /       2   \        2         /       2   \                                   \                                                   |   3/2     x   |                   |
 \/ x  |    /       2   \ |  /       2   \     3             /       2   \     2                            2         /       2   \                    2            2                       3         /       2   \                2         /       2   \                   |                \x        x          x    /          9*\1 + tan (x)/*\cos (tan(x))*\1 + tan (x)/ - 2*sin (tan(x))*\1 + tan (x)/ + 2*cos(tan(x))*sin(tan(x))*tan(x)/*cos(tan(x))*log(3)                                \  x            /                   |
3     *|- 3*\1 + tan (x)/*\2*\1 + tan (x)/ *sin (tan(x)) - 7*\1 + tan (x)/ *cos (tan(x))*sin(tan(x)) + 2*cos (tan(x))*\1 + tan (x)/*sin(tan(x)) + 4*cos (tan(x))*tan (x)*sin(tan(x)) + 6*cos (tan(x))*\1 + tan (x)/*tan(x) - 12*sin (tan(x))*\1 + tan (x)/*cos(tan(x))*tan(x)/ + ----------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------|
       |                                                                                                                                                                                                                                                                                                8                                                                                           ___                                                                                                4                                |
       \                                                                                                                                                                                                                                                                                                                                                                                        2*\/ x                                                                                                                                  /

$$3^{\sqrt{x}} \left(- \frac{9 \left(\frac{\log{\left(3 \right)}}{x} - \frac{1}{x^{\frac{3}{2}}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(\tan{\left(x \right)} \right)} \cos^{2}{\left(\tan{\left(x \right)} \right)}}{4} - 3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin^{3}{\left(\tan{\left(x \right)} \right)} - 7 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\tan{\left(x \right)} \right)} \cos^{2}{\left(\tan{\left(x \right)} \right)} - 12 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(\tan{\left(x \right)} \right)} \cos{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan{\left(x \right)} \right)} \cos^{2}{\left(\tan{\left(x \right)} \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos^{3}{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)} + 4 \sin{\left(\tan{\left(x \right)} \right)} \cos^{2}{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)}\right) + \frac{\left(- \frac{3 \log{\left(3 \right)}}{x^{2}} + \frac{\log{\left(3 \right)}^{2}}{x^{\frac{3}{2}}} + \frac{3}{x^{\frac{5}{2}}}\right) \log{\left(3 \right)} \cos^{3}{\left(\tan{\left(x \right)} \right)}}{8} - \frac{9 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- 2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(\tan{\left(x \right)} \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(\tan{\left(x \right)} \right)} + 2 \sin{\left(\tan{\left(x \right)} \right)} \cos{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) \log{\left(3 \right)} \cos{\left(\tan{\left(x \right)} \right)}}{2 \sqrt{x}}\right)$$

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Gráfico

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Expresiones idénticas

Expresiones con funciones

Derivada de y=3^sqrt(x)cos^3(tgx)

Gráfico:

Definida a trozos:

Solución