Mister Exam

Derivative of ln(sin4x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(sin(4*x))
$$\log{\left(\sin{\left(4 x \right)} \right)}$$
log(sin(4*x))
Detail solution
  1. Let .

  2. The derivative of is .

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

    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:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
4*cos(4*x)
----------
 sin(4*x) 
$$\frac{4 \cos{\left(4 x \right)}}{\sin{\left(4 x \right)}}$$
The second derivative [src]
    /       2     \
    |    cos (4*x)|
-16*|1 + ---------|
    |       2     |
    \    sin (4*x)/
$$- 16 \left(1 + \frac{\cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right)$$
The third derivative [src]
    /       2     \         
    |    cos (4*x)|         
128*|1 + ---------|*cos(4*x)
    |       2     |         
    \    sin (4*x)/         
----------------------------
          sin(4*x)          
$$\frac{128 \left(1 + \frac{\cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \cos{\left(4 x \right)}}{\sin{\left(4 x \right)}}$$