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Derivative of ch(x)
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Derivative of tan^3(x^2)
Identical expressions
tan^ three (x^ two)
tangent of cubed (x squared )
tangent of to the power of three (x to the power of two)
tan3(x2)
tan3x2
tan³(x²)
tan to the power of 3(x to the power of 2)
tan^3x^2

The derivative of the function/ tan^3(x^2)

Derivative of tan^3(x^2)

The solution

You have entered [src]

   3/ 2\
tan \x /

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

d /   3/ 2\\
--\tan \x //
dx

$$\frac{d}{d x} \tan^{3}{\left(x^{2} \right)}$$

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2/ 2\ /       2/ 2\\
6*x*tan \x /*\1 + tan \x //

$$6 x \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \tan^{2}{\left(x^{2} \right)}$$

Simplify

The second derivative [src]

  /       2/ 2\\ /   2    2/ 2\      2 /       2/ 2\\      / 2\\    / 2\
6*\1 + tan \x //*\4*x *tan \x / + 4*x *\1 + tan \x // + tan\x //*tan\x /

$$6 \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \left(4 x^{2} \tan^{2}{\left(x^{2} \right)} + 4 x^{2} \left(\tan^{2}{\left(x^{2} \right)} + 1\right) + \tan{\left(x^{2} \right)}\right) \tan{\left(x^{2} \right)}$$

Simplify

The third derivative [src]

                    /                                2                                                                           \
     /       2/ 2\\ |     3/ 2\      2 /       2/ 2\\      /       2/ 2\\    / 2\      2    4/ 2\       2    2/ 2\ /       2/ 2\\|
24*x*\1 + tan \x //*\3*tan \x / + 2*x *\1 + tan \x //  + 3*\1 + tan \x //*tan\x / + 4*x *tan \x / + 14*x *tan \x /*\1 + tan \x ///

$$24 x \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \left(4 x^{2} \tan^{4}{\left(x^{2} \right)} + 14 x^{2} \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \tan^{2}{\left(x^{2} \right)} + 2 x^{2} \left(\tan^{2}{\left(x^{2} \right)} + 1\right)^{2} + 3 \tan^{3}{\left(x^{2} \right)} + 3 \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \tan{\left(x^{2} \right)}\right)$$

Simplify

The graph

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Identical expressions

Derivative of tan^3(x^2)

The graph:

Piecewise:

The solution