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y=ln(tgx^2)

Derivative of y=ln(tgx^2)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   2   \
log\tan (x)/
$$\log{\left(\tan^{2}{\left(x \right)} \right)}$$
log(tan(x)^2)
Detail solution
  1. Let .

  2. The derivative of is .

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Rewrite the function to be differentiated:

      2. Apply the quotient rule, which is:

        and .

        To find :

        1. The derivative of sine is cosine:

        To find :

        1. The derivative of cosine is negative sine:

        Now plug in to the quotient rule:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
         2   
2 + 2*tan (x)
-------------
    tan(x)   
$$\frac{2 \tan^{2}{\left(x \right)} + 2}{\tan{\left(x \right)}}$$
The second derivative [src]
  /                             2\
  |                /       2   \ |
  |         2      \1 + tan (x)/ |
2*|2 + 2*tan (x) - --------------|
  |                      2       |
  \                   tan (x)    /
$$2 \left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2\right)$$
The third derivative [src]
                /                        2                  \
                |           /       2   \      /       2   \|
  /       2   \ |           \1 + tan (x)/    2*\1 + tan (x)/|
4*\1 + tan (x)/*|2*tan(x) + -------------- - ---------------|
                |                 3               tan(x)    |
                \              tan (x)                      /
$$4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + 2 \tan{\left(x \right)}\right)$$
The graph
Derivative of y=ln(tgx^2)