Mister Exam

Derivative of log(tan4x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(tan(4*x))
$$\log{\left(\tan{\left(4 x \right)} \right)}$$
d                
--(log(tan(4*x)))
dx               
$$\frac{d}{d x} \log{\left(\tan{\left(4 x \right)} \right)}$$
Detail solution
  1. Let .

  2. The derivative of is .

  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. Let .

      2. The derivative of sine is cosine:

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

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      To find :

      1. Let .

      2. The derivative of cosine is negative sine:

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

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      Now plug in to the quotient rule:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
         2     
4 + 4*tan (4*x)
---------------
    tan(4*x)   
$$\frac{4 \tan^{2}{\left(4 x \right)} + 4}{\tan{\left(4 x \right)}}$$
The second derivative [src]
   /                                 2\
   |                  /       2     \ |
   |         2        \1 + tan (4*x)/ |
16*|2 + 2*tan (4*x) - ----------------|
   |                        2         |
   \                     tan (4*x)    /
$$16 \left(- \frac{\left(\tan^{2}{\left(4 x \right)} + 1\right)^{2}}{\tan^{2}{\left(4 x \right)}} + 2 \tan^{2}{\left(4 x \right)} + 2\right)$$
The third derivative [src]
                    /                            2                    \
                    |             /       2     \      /       2     \|
    /       2     \ |             \1 + tan (4*x)/    2*\1 + tan (4*x)/|
128*\1 + tan (4*x)/*|2*tan(4*x) + ---------------- - -----------------|
                    |                   3                 tan(4*x)    |
                    \                tan (4*x)                        /
$$128 \left(\tan^{2}{\left(4 x \right)} + 1\right) \left(\frac{\left(\tan^{2}{\left(4 x \right)} + 1\right)^{2}}{\tan^{3}{\left(4 x \right)}} - \frac{2 \left(\tan^{2}{\left(4 x \right)} + 1\right)}{\tan{\left(4 x \right)}} + 2 \tan{\left(4 x \right)}\right)$$
The graph
Derivative of log(tan4x)