Derivative of tg^5(x)

The solution

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   5   
tan (x)

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

d /   5   \
--\tan (x)/
dx

$$\frac{d}{d x} \tan^{5}{\left(x \right)}$$

Transform

Detail solution

Let $u = \tan{\left(x \right)}$ .
Apply the power rule: $u^{5}$ goes to $5 u^{4}$
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 $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. 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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   4    /         2   \
tan (x)*\5 + 5*tan (x)/

$$\left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)}$$

Simplify

The second derivative [src]

      3    /       2   \ /         2   \
10*tan (x)*\1 + tan (x)/*\2 + 3*tan (x)/

$$10 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 2\right) \tan^{3}{\left(x \right)}$$

Simplify

The third derivative [src]

                         /                           2                           \
      2    /       2   \ |     4        /       2   \          2    /       2   \|
10*tan (x)*\1 + tan (x)/*\2*tan (x) + 6*\1 + tan (x)/  + 13*tan (x)*\1 + tan (x)//

$$10 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{4}{\left(x \right)} + 13 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}\right) \tan^{2}{\left(x \right)}$$

Simplify

The graph

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Derivative of tg^5(x)

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Piecewise:

The solution