Derivative of sqrt(tan5x)

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  __________
\/ tan(5*x)

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

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

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

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Detail solution

Let $u = \tan{\left(5 x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   /       2     \ /                        2     \
   |1   tan (5*x)| |    __________   1 + tan (5*x)|
25*|- + ---------|*|4*\/ tan(5*x)  - -------------|
   \4       4    / |                     3/2      |
                   \                  tan   (5*x) /

$$25 \left(- \frac{\tan^{2}{\left(5 x \right)} + 1}{\tan^{\frac{3}{2}}{\left(5 x \right)}} + 4 \sqrt{\tan{\left(5 x \right)}}\right) \left(\frac{\tan^{2}{\left(5 x \right)}}{4} + \frac{1}{4}\right)$$

Simplify

The third derivative [src]

                    /                                                      2\
    /       2     \ |                   /       2     \     /       2     \ |
    |1   tan (5*x)| |      3/2        4*\1 + tan (5*x)/   3*\1 + tan (5*x)/ |
125*|- + ---------|*|16*tan   (5*x) - ----------------- + ------------------|
    \8       8    / |                      __________           5/2         |
                    \                    \/ tan(5*x)         tan   (5*x)    /

$$125 \left(\frac{\tan^{2}{\left(5 x \right)}}{8} + \frac{1}{8}\right) \left(\frac{3 \left(\tan^{2}{\left(5 x \right)} + 1\right)^{2}}{\tan^{\frac{5}{2}}{\left(5 x \right)}} - \frac{4 \left(\tan^{2}{\left(5 x \right)} + 1\right)}{\sqrt{\tan{\left(5 x \right)}}} + 16 \tan^{\frac{3}{2}}{\left(5 x \right)}\right)$$

Simplify

The graph

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Identical expressions

Derivative of sqrt(tan5x)

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The solution