Derivative of sqrt(tan(x))

The solution

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  ________
\/ tan(x)

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

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

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

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Detail solution

Let $u = \tan{\left(x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
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:

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

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

The answer is:

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

The first derivative [src]

       2   
1   tan (x)
- + -------
2      2   
-----------
   ________
 \/ tan(x)

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

Simplify

The second derivative [src]

/       2   \ /                      2   \
|1   tan (x)| |    ________   1 + tan (x)|
|- + -------|*|4*\/ tan(x)  - -----------|
\4      4   / |                   3/2    |
              \                tan   (x) /

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of sqrt(tan(x))

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