Derivative of y=sin(tan(sqrt(x)))

The solution

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   /   /  ___\\
sin\tan\\/ x //

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

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

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

Transform

Detail solution

Let $u = \tan{\left(\sqrt{x} \right)}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\sqrt{x} \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(\sqrt{x} \right)} = \frac{\sin{\left(\sqrt{x} \right)}}{\cos{\left(\sqrt{x} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(\sqrt{x} \right)}$ 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:
  
  $\frac{\frac{\sin^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}} + \frac{\cos^{2}{\left(\sqrt{x} \right)}}{2 \sqrt{x}}}{\cos^{2}{\left(\sqrt{x} \right)}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of y=sin(tan(sqrt(x)))

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