Derivada de y=sin(x)cos(x)tan(x)

Solución

Ha introducido [src]

sin(x)*cos(x)*tan(x)

$$\sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)}$$

(sin(x)*cos(x))*tan(x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = \sin{\left(x \right)} \cos{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. La derivada del seno es igual al coseno:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  $g{\left(x \right)} = \cos{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. La derivada del coseno es igual a menos el seno:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Como resultado de: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$
$g{\left(x \right)} = \tan{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
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: $\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}}$
Simplificamos:

$\sin{\left(2 x \right)}$

Respuesta:

$\sin{\left(2 x \right)}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

$$\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplificar

Segunda derivada [src]

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

$$2 \left(- \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)}\right)$$

Simplificar

Tercera derivada [src]

  /  /   2         2   \            /       2   \                   /       2   \ /   2         2   \          /       2   \ /         2   \              \
2*\2*\sin (x) - cos (x)/*tan(x) - 6*\1 + tan (x)/*cos(x)*sin(x) - 3*\1 + tan (x)/*\sin (x) - cos (x)/*tan(x) + \1 + tan (x)/*\1 + 3*tan (x)/*cos(x)*sin(x)/

$$2 \left(- 3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}\right)$$

Simplificar

Gráfico

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Derivada de y=sin(x)cos(x)tan(x)

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