Derivative of y=tg2x^5

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   5     
tan (2*x)

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

d /   5     \
--\tan (2*x)/
dx

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

Transform

Detail solution

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

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

$\frac{10 \tan^{4}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$

The answer is:

$\frac{10 \tan^{4}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   4      /           2     \
tan (2*x)*\10 + 10*tan (2*x)/

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

Simplify

The second derivative [src]

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

$$40 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(3 \tan^{2}{\left(2 x \right)} + 2\right) \tan^{3}{\left(2 x \right)}$$

Simplify

The third derivative [src]

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

$$80 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(6 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} + 13 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan^{2}{\left(2 x \right)} + 2 \tan^{4}{\left(2 x \right)}\right) \tan^{2}{\left(2 x \right)}$$

Simplify

The graph

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Derivative of y=tg2x^5

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The solution