Derivative of logtan(x²)

The solution

You have entered [src]

   /   / 2\\
log\tan\x //

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

log(tan(x^2))

Detail solution

Let $u = \tan{\left(x^{2} \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x^{2} \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(x^{2} \right)} = \frac{\sin{\left(x^{2} \right)}}{\cos{\left(x^{2} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x^{2} \right)}$ and $g{\left(x \right)} = \cos{\left(x^{2} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x^{2}$ .
  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 $\frac{d}{d x} x^{2}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    The result of the chain rule is:
    
    $2 x \cos{\left(x^{2} \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = x^{2}$ .
  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 $\frac{d}{d x} x^{2}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    The result of the chain rule is:
    
    $- 2 x \sin{\left(x^{2} \right)}$
  Now plug in to the quotient rule:
  
  $\frac{2 x \sin^{2}{\left(x^{2} \right)} + 2 x \cos^{2}{\left(x^{2} \right)}}{\cos^{2}{\left(x^{2} \right)}}$
The result of the chain rule is:

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

$\frac{4 x}{\sin{\left(2 x^{2} \right)}}$

The answer is:

$\frac{4 x}{\sin{\left(2 x^{2} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of logtan(x²)

The graph:

Piecewise:

The solution