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Derivative of:
Derivative of sqrt(cos(x))
Derivative of cos(2t)
Derivative of 5*tan(x)
Derivative of 3xsinx
Identical expressions
y=sin(cos^ two (tg^3x))
y equally sinus of ( co sinus of e of squared (tg cubed x))
y equally sinus of ( co sinus of e of to the power of two (tg cubed x))
y=sin(cos2(tg3x))
y=sincos2tg3x
y=sin(cos²(tg³x))
y=sin(cos to the power of 2(tg to the power of 3x))
y=sincos^2tg^3x

The derivative of the function/ y=sin(cos^2(tg^3x))

Derivative of y=sin(cos^2(tg^3x))

The solution

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   /   2/   3   \\
sin\cos \tan (x)//

$$\sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)}$$

sin(cos(tan(x)^3)^2)

Detail solution

Let $u = \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)}$ :
1. Let $u = \cos{\left(\tan^{3}{\left(x \right)} \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(\tan^{3}{\left(x \right)} \right)}$ :
  1. Let $u = \tan^{3}{\left(x \right)}$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan^{3}{\left(x \right)}$ :
    1. Let $u = \tan{\left(x \right)}$ .
    2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
      1. Rewrite the function to be differentiated:
        
        $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
      2. Apply the quotient rule, which is:
        
        $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
        
        $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
        
        To find $\frac{d}{d x} f{\left(x \right)}$ :
        
        The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        To find $\frac{d}{d x} g{\left(x \right)}$ :
        
        The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
        
        Now plug in to the quotient rule:
        
        $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
      The result of the chain rule is:
      
      $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    The result of the chain rule is:
    
    $- \frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $- \frac{6 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result of the chain rule is:

$- \frac{6 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                 /               2                                                               2                                                                                                                                                                               2                                                                                                                       2                                                                         2                                                                                                                                                                                                                           2                                                                          2                                                                        2                                                      \
   /       2   \ |  /       2   \     /   2/   3   \\    /   3   \    /   3   \     /       2   \     2/   3   \    3       /   2/   3   \\        2/   3   \    5    /       2   \    /   2/   3   \\        4       /   2/   3   \\    /   3   \    /   3   \     /       2   \     2/   3   \    3       /   2/   3   \\        2/   3   \    5    /       2   \    /   2/   3   \\      /       2   \     3/   3   \    6       /   2/   3   \\    /   3   \      /       2   \     2/   3   \    2/   3   \    3       /   2/   3   \\         2/   3   \    2/   3   \    5    /       2   \    /   2/   3   \\        2    /       2   \    /   2/   3   \\    /   3   \    /   3   \      /       2   \     3/   3   \    3/   3   \    6       /   2/   3   \\      /       2   \     6       /   2/   3   \\    /   3   \    /   3   \      /       2   \     3/   3   \    6       /   3   \    /   2/   3   \\|
12*\1 + tan (x)/*\- \1 + tan (x)/ *cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ - 9*\1 + tan (x)/ *cos \tan (x)/*tan (x)*cos\cos \tan (x)// - 9*cos \tan (x)/*tan (x)*\1 + tan (x)/*cos\cos \tan (x)// - 2*tan (x)*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ + 9*\1 + tan (x)/ *sin \tan (x)/*tan (x)*cos\cos \tan (x)// + 9*sin \tan (x)/*tan (x)*\1 + tan (x)/*cos\cos \tan (x)// - 27*\1 + tan (x)/ *cos \tan (x)/*tan (x)*sin\cos \tan (x)//*sin\tan (x)/ - 18*\1 + tan (x)/ *cos \tan (x)/*sin \tan (x)/*tan (x)*sin\cos \tan (x)// - 18*cos \tan (x)/*sin \tan (x)/*tan (x)*\1 + tan (x)/*sin\cos \tan (x)// - 7*tan (x)*\1 + tan (x)/*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ + 18*\1 + tan (x)/ *cos \tan (x)/*sin \tan (x)/*tan (x)*cos\cos \tan (x)// + 18*\1 + tan (x)/ *tan (x)*cos\cos \tan (x)//*cos\tan (x)/*sin\tan (x)/ + 27*\1 + tan (x)/ *sin \tan (x)/*tan (x)*cos\tan (x)/*sin\cos \tan (x)///

$$12 \left(\tan^{2}{\left(x \right)} + 1\right) \left(27 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin^{3}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} - 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} - 27 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos^{3}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} + 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin^{3}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos^{3}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} + 9 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \tan^{3}{\left(x \right)} + 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{6}{\left(x \right)} - \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} - 9 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} - 18 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{5}{\left(x \right)} + 9 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \tan^{5}{\left(x \right)} - 7 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{2}{\left(x \right)} - 9 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \tan^{5}{\left(x \right)} - 2 \sin{\left(\tan^{3}{\left(x \right)} \right)} \cos{\left(\cos^{2}{\left(\tan^{3}{\left(x \right)} \right)} \right)} \cos{\left(\tan^{3}{\left(x \right)} \right)} \tan^{4}{\left(x \right)}\right)$$

Simplify

The graph

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How to use it?

Identical expressions

Derivative of y=sin(cos^2(tg^3x))

The graph:

Piecewise:

The solution