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Derivative of:
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Identical expressions
sqrttan(x)^(two)
square root of tangent of (x) to the power of (2)
square root of tangent of (x) to the power of (two)
√tan(x)^(2)
sqrttan(x)(2)
sqrttanx2
sqrttanx^2

The derivative of the function/ sqrttan(x)^(2)

Derivative of sqrttan(x)^(2)

The solution

You have entered [src]

          2
  ________ 
\/ tan(x)

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

  /          2\
d |  ________ |
--\\/ tan(x)  /
dx

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

Transform

Detail solution

Let $u = \sqrt{\tan{\left(x \right)}}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{\tan{\left(x \right)}}$ :
1. Let $u = \tan{\left(x \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. 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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of sqrttan(x)^(2)

The graph:

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The solution