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Derivada de:
Derivada de x^-7
Derivada de i*n*x
Derivada de 5*tan(x)^3+sin(4*x)*e^((-x)/7)
Derivada de (-2)/x^3
Expresiones idénticas
y=cos^ cinco ⁡(tan⁡(sin⁡(x^ cinco)))
y es igual a coseno de en el grado 5⁡( tangente de ⁡( seno de ⁡(x en el grado 5)))
y es igual a coseno de en el grado cinco ⁡( tangente de ⁡( seno de ⁡(x en el grado cinco)))
y=cos5⁡(tan⁡(sin⁡(x5)))
y=cos5⁡tan⁡sin⁡x5
y=cos⁵⁡(tan⁡(sin⁡(x⁵)))
y=cos^5⁡tan⁡sin⁡x^5
Expresiones con funciones
Coseno cos
cos(x)+49
cos(x-1)
cos(sqrt(x)+3*x)
cos(x)*x^3
cos(asin(x))
Tangente tan
tan(x^2)
tan(0)
tan(x^2+1)^(4)
tan(2*p)/(1-cot(2*p))
tan^-1√x
Seno sin
sin(x)-x
sin(x)/(x*cos(x))
sin(x)/log(x)
sin(x+(pi/4))
sin(x)/(x^2+1)

Derivada de la función/ (x^5)/ y=cos/ y=cos^5⁡(tan⁡(sin⁡(x^5)))

Derivada de y=cos^5⁡(tan⁡(sin⁡(x^5)))

Solución

Ha introducido [src]

   5/   /   / 5\\\
cos \tan\sin\x ///

$$\cos^{5}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}$$

cos(tan(sin(x^5)))^5

Solución detallada

Sustituimos $u = \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}$ .
Según el principio, aplicamos: $u^{5}$ tenemos $5 u^{4}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}$ :
1. Sustituimos $u = \tan{\left(\sin{\left(x^{5} \right)} \right)}$ .
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 $\frac{d}{d x} \tan{\left(\sin{\left(x^{5} \right)} \right)}$ :
  1. Reescribimos las funciones para diferenciar:
    
    $\tan{\left(\sin{\left(x^{5} \right)} \right)} = \frac{\sin{\left(\sin{\left(x^{5} \right)} \right)}}{\cos{\left(\sin{\left(x^{5} \right)} \right)}}$
  2. Se aplica la regla de la derivada parcial:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(\sin{\left(x^{5} \right)} \right)}$ y $g{\left(x \right)} = \cos{\left(\sin{\left(x^{5} \right)} \right)}$ .
    
    Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
    1. Sustituimos $u = \sin{\left(x^{5} \right)}$ .
    2. La derivada del seno es igual al coseno:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(x^{5} \right)}$ :
      1. Sustituimos $u = x^{5}$ .
      2. La derivada del seno es igual al coseno:
        
        $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} x^{5}$ :
        
        Según el principio, aplicamos: $x^{5}$ tenemos $5 x^{4}$
        
        Como resultado de la secuencia de reglas:
        
        $5 x^{4} \cos{\left(x^{5} \right)}$
      Como resultado de la secuencia de reglas:
      
      $5 x^{4} \cos{\left(x^{5} \right)} \cos{\left(\sin{\left(x^{5} \right)} \right)}$
    Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
    1. Sustituimos $u = \sin{\left(x^{5} \right)}$ .
    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 $\frac{d}{d x} \sin{\left(x^{5} \right)}$ :
      1. Sustituimos $u = x^{5}$ .
      2. La derivada del seno es igual al coseno:
        
        $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} x^{5}$ :
        
        Según el principio, aplicamos: $x^{5}$ tenemos $5 x^{4}$
        
        Como resultado de la secuencia de reglas:
        
        $5 x^{4} \cos{\left(x^{5} \right)}$
      Como resultado de la secuencia de reglas:
      
      $- 5 x^{4} \sin{\left(\sin{\left(x^{5} \right)} \right)} \cos{\left(x^{5} \right)}$
    Ahora aplicamos la regla de la derivada de una divesión:
    
    $\frac{5 x^{4} \sin^{2}{\left(\sin{\left(x^{5} \right)} \right)} \cos{\left(x^{5} \right)} + 5 x^{4} \cos{\left(x^{5} \right)} \cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}{\cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}$
  Como resultado de la secuencia de reglas:
  
  $- \frac{\left(5 x^{4} \sin^{2}{\left(\sin{\left(x^{5} \right)} \right)} \cos{\left(x^{5} \right)} + 5 x^{4} \cos{\left(x^{5} \right)} \cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}\right) \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}}{\cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}$
Como resultado de la secuencia de reglas:

$- \frac{5 \left(5 x^{4} \sin^{2}{\left(\sin{\left(x^{5} \right)} \right)} \cos{\left(x^{5} \right)} + 5 x^{4} \cos{\left(x^{5} \right)} \cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}\right) \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{4}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}}{\cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}$
Simplificamos:

$- \frac{25 x^{4} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{4}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}}{\cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}$

Respuesta:

$- \frac{25 x^{4} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{4}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}}{\cos^{2}{\left(\sin{\left(x^{5} \right)} \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

     4    4/   /   / 5\\\ /       2/   / 5\\\    / 5\    /   /   / 5\\\
-25*x *cos \tan\sin\x ///*\1 + tan \sin\x ///*cos\x /*sin\tan\sin\x ///

$$- 25 x^{4} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{4}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}$$

Simplificar

Segunda derivada [src]

    3    3/   /   / 5\\\ /       2/   / 5\\\ /       / 5\    /   /   / 5\\\    /   /   / 5\\\      5    2/ 5\    2/   /   / 5\\\ /       2/   / 5\\\      5    /   /   / 5\\\    / 5\    /   /   / 5\\\       5    2/ 5\    2/   /   / 5\\\ /       2/   / 5\\\       5    2/ 5\    /   /   / 5\\\    /   /   / 5\\\    /   / 5\\\
25*x *cos \tan\sin\x ///*\1 + tan \sin\x ///*\- 4*cos\x /*cos\tan\sin\x ///*sin\tan\sin\x /// - 5*x *cos \x /*cos \tan\sin\x ///*\1 + tan \sin\x /// + 5*x *cos\tan\sin\x ///*sin\x /*sin\tan\sin\x /// + 20*x *cos \x /*sin \tan\sin\x ///*\1 + tan \sin\x /// - 10*x *cos \x /*cos\tan\sin\x ///*sin\tan\sin\x ///*tan\sin\x ///

$$25 x^{3} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \left(20 x^{5} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{2}{\left(x^{5} \right)} - 5 x^{5} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \cos^{2}{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} + 5 x^{5} \sin{\left(x^{5} \right)} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} - 10 x^{5} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{2}{\left(x^{5} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan{\left(\sin{\left(x^{5} \right)} \right)} - 4 \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}\right) \cos^{3}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}$$

Simplificar

Tercera derivada [src]

                                             /                                                                               2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    2                                                                                                                                                                                                                                                                                                      \
    2    2/   /   / 5\\\ /       2/   / 5\\\ |        2/   /   / 5\\\    / 5\    /   /   / 5\\\        10 /       2/   / 5\\\     3/ 5\    3/   /   / 5\\\       5    2/ 5\    3/   /   / 5\\\ /       2/   / 5\\\       10    2/   /   / 5\\\    / 5\    /   /   / 5\\\       5    2/   /   / 5\\\    / 5\    /   /   / 5\\\        10    3/ 5\    3/   /   / 5\\\ /       2/   / 5\\\    /   / 5\\        5    2/ 5\    2/   /   / 5\\\    /   /   / 5\\\    /   / 5\\        10    3/ 5\    2/   /   / 5\\\    2/   / 5\\    /   /   / 5\\\       10    3/ 5\    2/   /   / 5\\\ /       2/   / 5\\\    /   /   / 5\\\       10    3/   /   / 5\\\ /       2/   / 5\\\    / 5\    / 5\        5    2/ 5\    2/   /   / 5\\\ /       2/   / 5\\\    /   /   / 5\\\        10 /       2/   / 5\\\     3/ 5\    2/   /   / 5\\\    /   /   / 5\\\        10    2/   /   / 5\\\ /       2/   / 5\\\    / 5\    /   /   / 5\\\    / 5\        10    2/   /   / 5\\\    / 5\    / 5\    /   /   / 5\\\    /   / 5\\        10    3/ 5\    2/   /   / 5\\\ /       2/   / 5\\\    /   /   / 5\\\    /   / 5\\|
25*x *cos \tan\sin\x ///*\1 + tan \sin\x ///*\- 12*cos \tan\sin\x ///*cos\x /*sin\tan\sin\x /// - 300*x  *\1 + tan \sin\x /// *cos \x /*sin \tan\sin\x /// - 60*x *cos \x /*cos \tan\sin\x ///*\1 + tan \sin\x /// + 25*x  *cos \tan\sin\x ///*cos\x /*sin\tan\sin\x /// + 60*x *cos \tan\sin\x ///*sin\x /*sin\tan\sin\x /// - 150*x  *cos \x /*cos \tan\sin\x ///*\1 + tan \sin\x ///*tan\sin\x // - 120*x *cos \x /*cos \tan\sin\x ///*sin\tan\sin\x ///*tan\sin\x // - 100*x  *cos \x /*cos \tan\sin\x ///*tan \sin\x //*sin\tan\sin\x /// - 50*x  *cos \x /*cos \tan\sin\x ///*\1 + tan \sin\x ///*sin\tan\sin\x /// + 75*x  *cos \tan\sin\x ///*\1 + tan \sin\x ///*cos\x /*sin\x / + 240*x *cos \x /*sin \tan\sin\x ///*\1 + tan \sin\x ///*cos\tan\sin\x /// + 325*x  *\1 + tan \sin\x /// *cos \x /*cos \tan\sin\x ///*sin\tan\sin\x /// - 300*x  *sin \tan\sin\x ///*\1 + tan \sin\x ///*cos\x /*cos\tan\sin\x ///*sin\x / + 150*x  *cos \tan\sin\x ///*cos\x /*sin\x /*sin\tan\sin\x ///*tan\sin\x // + 600*x  *cos \x /*sin \tan\sin\x ///*\1 + tan \sin\x ///*cos\tan\sin\x ///*tan\sin\x ///

$$25 x^{2} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \left(- 300 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right)^{2} \sin^{3}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{3}{\left(x^{5} \right)} + 325 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right)^{2} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{3}{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} - 300 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin{\left(x^{5} \right)} \sin^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} + 75 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin{\left(x^{5} \right)} \cos{\left(x^{5} \right)} \cos^{3}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} + 600 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{3}{\left(x^{5} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan{\left(\sin{\left(x^{5} \right)} \right)} - 50 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{3}{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} - 150 x^{10} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \cos^{3}{\left(x^{5} \right)} \cos^{3}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan{\left(\sin{\left(x^{5} \right)} \right)} + 150 x^{10} \sin{\left(x^{5} \right)} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan{\left(\sin{\left(x^{5} \right)} \right)} - 100 x^{10} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{3}{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 25 x^{10} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} + 240 x^{5} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \sin^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{2}{\left(x^{5} \right)} \cos{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} - 60 x^{5} \left(\tan^{2}{\left(\sin{\left(x^{5} \right)} \right)} + 1\right) \cos^{2}{\left(x^{5} \right)} \cos^{3}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} + 60 x^{5} \sin{\left(x^{5} \right)} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} - 120 x^{5} \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos^{2}{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \tan{\left(\sin{\left(x^{5} \right)} \right)} - 12 \sin{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)} \cos{\left(x^{5} \right)} \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}\right) \cos^{2}{\left(\tan{\left(\sin{\left(x^{5} \right)} \right)} \right)}$$

Simplificar

Gráfico

Otras calculadoras

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Expresiones idénticas

Expresiones con funciones

Derivada de y=cos^5⁡(tan⁡(sin⁡(x^5)))

Gráfico:

Definida a trozos:

Solución