Mister Exam

Derivative of ln(tan(3x))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(tan(3*x))
$$\log{\left(\tan{\left(3 x \right)} \right)}$$
d                
--(log(tan(3*x)))
dx               
$$\frac{d}{d x} \log{\left(\tan{\left(3 x \right)} \right)}$$
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     
3 + 3*tan (3*x)
---------------
    tan(3*x)   
$$\frac{3 \tan^{2}{\left(3 x \right)} + 3}{\tan{\left(3 x \right)}}$$
The second derivative [src]
  /                                 2\
  |                  /       2     \ |
  |         2        \1 + tan (3*x)/ |
9*|2 + 2*tan (3*x) - ----------------|
  |                        2         |
  \                     tan (3*x)    /
$$9 \left(- \frac{\left(\tan^{2}{\left(3 x \right)} + 1\right)^{2}}{\tan^{2}{\left(3 x \right)}} + 2 \tan^{2}{\left(3 x \right)} + 2\right)$$
The third derivative [src]
                   /                            2                    \
                   |             /       2     \      /       2     \|
   /       2     \ |             \1 + tan (3*x)/    2*\1 + tan (3*x)/|
54*\1 + tan (3*x)/*|2*tan(3*x) + ---------------- - -----------------|
                   |                   3                 tan(3*x)    |
                   \                tan (3*x)                        /
$$54 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(\frac{\left(\tan^{2}{\left(3 x \right)} + 1\right)^{2}}{\tan^{3}{\left(3 x \right)}} - \frac{2 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\tan{\left(3 x \right)}} + 2 \tan{\left(3 x \right)}\right)$$
The graph
Derivative of ln(tan(3x))