Derivative of tan(sqrt(x))

The solution

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   /  ___\
tan\\/ x /

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

d /   /  ___\\
--\tan\\/ x //
dx

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

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Detail solution

Rewrite the function to be differentiated:

$\tan{\left(\sqrt{x} \right)} = \frac{\sin{\left(\sqrt{x} \right)}}{\cos{\left(\sqrt{x} \right)}}$
Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(\sqrt{x} \right)}$ and $g{\left(x \right)} = \cos{\left(\sqrt{x} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sqrt{x}$ .
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 $\frac{d}{d x} \sqrt{x}$ :
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  The result of the chain rule is:
  
  $\frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sqrt{x}$ .
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 $\frac{d}{d x} \sqrt{x}$ :
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The first derivative [src]

       2/  ___\
1 + tan \\/ x /
---------------
        ___    
    2*\/ x

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

Simplify

The second derivative [src]

                  /              /  ___\\
/       2/  ___\\ |   1     2*tan\\/ x /|
\1 + tan \\/ x //*|- ---- + ------------|
                  |   3/2        x      |
                  \  x                  /
-----------------------------------------
                    4

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

Simplify

The third derivative [src]

                  /            /  ___\     /       2/  ___\\        2/  ___\\
/       2/  ___\\ | 3     6*tan\\/ x /   2*\1 + tan \\/ x //   4*tan \\/ x /|
\1 + tan \\/ x //*|---- - ------------ + ------------------- + -------------|
                  | 5/2         2                 3/2                3/2    |
                  \x           x                 x                  x       /
-----------------------------------------------------------------------------
                                      8

$$\frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \left(- \frac{6 \tan{\left(\sqrt{x} \right)}}{x^{2}} + \frac{2 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)}{x^{\frac{3}{2}}} + \frac{4 \tan^{2}{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}} + \frac{3}{x^{\frac{5}{2}}}\right)}{8}$$

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Derivative of tan(sqrt(x))

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