Derivative of tg5x^1/5

The solution

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5 __________
\/ tan(5*x)

$$\sqrt[5]{\tan{\left(5 x \right)}}$$

tan(5*x)^(1/5)

Detail solution

Let $u = \tan{\left(5 x \right)}$ .
Apply the power rule: $\sqrt[5]{u}$ goes to $\frac{1}{5 u^{\frac{4}{5}}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(5 x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(5 x \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(5 x \right)}$ and $g{\left(x \right)} = \cos{\left(5 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $5$
    The result of the chain rule is:
    
    $5 \cos{\left(5 x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $5$
    The result of the chain rule is:
    
    $- 5 \sin{\left(5 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}$
The result of the chain rule is:

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

$\frac{1}{\cos^{2}{\left(5 x \right)} \tan^{\frac{4}{5}}{\left(5 x \right)}}$

The answer is:

$\frac{1}{\cos^{2}{\left(5 x \right)} \tan^{\frac{4}{5}}{\left(5 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2     
1 + tan (5*x)
-------------
    4/5      
 tan   (5*x)

$$\frac{\tan^{2}{\left(5 x \right)} + 1}{\tan^{\frac{4}{5}}{\left(5 x \right)}}$$

Simplify

The second derivative [src]

                  /                   /       2     \\
  /       2     \ |  5 __________   2*\1 + tan (5*x)/|
2*\1 + tan (5*x)/*|5*\/ tan(5*x)  - -----------------|
                  |                       9/5        |
                  \                    tan   (5*x)   /

$$2 \left(- \frac{2 \left(\tan^{2}{\left(5 x \right)} + 1\right)}{\tan^{\frac{9}{5}}{\left(5 x \right)}} + 5 \sqrt[5]{\tan{\left(5 x \right)}}\right) \left(\tan^{2}{\left(5 x \right)} + 1\right)$$

Simplify

The third derivative [src]

                  /                                                        2\
                  |                    /       2     \      /       2     \ |
  /       2     \ |      6/5        35*\1 + tan (5*x)/   18*\1 + tan (5*x)/ |
2*\1 + tan (5*x)/*|50*tan   (5*x) - ------------------ + -------------------|
                  |                       4/5                   14/5        |
                  \                    tan   (5*x)           tan    (5*x)   /

$$2 \left(\tan^{2}{\left(5 x \right)} + 1\right) \left(\frac{18 \left(\tan^{2}{\left(5 x \right)} + 1\right)^{2}}{\tan^{\frac{14}{5}}{\left(5 x \right)}} - \frac{35 \left(\tan^{2}{\left(5 x \right)} + 1\right)}{\tan^{\frac{4}{5}}{\left(5 x \right)}} + 50 \tan^{\frac{6}{5}}{\left(5 x \right)}\right)$$

Simplify

The graph

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Derivative of tg5x^1/5

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The solution