Mister Exam

Derivative of y=ln(sin(3x-1))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(sin(3*x - 1))
$$\log{\left(\sin{\left(3 x - 1 \right)} \right)}$$
log(sin(3*x - 1))
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. Differentiate term by term:

        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:

        2. The derivative of the constant is zero.

        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]
3*cos(3*x - 1)
--------------
 sin(3*x - 1) 
$$\frac{3 \cos{\left(3 x - 1 \right)}}{\sin{\left(3 x - 1 \right)}}$$
The second derivative [src]
   /       2          \
   |    cos (-1 + 3*x)|
-9*|1 + --------------|
   |       2          |
   \    sin (-1 + 3*x)/
$$- 9 \left(1 + \frac{\cos^{2}{\left(3 x - 1 \right)}}{\sin^{2}{\left(3 x - 1 \right)}}\right)$$
The third derivative [src]
   /       2          \              
   |    cos (-1 + 3*x)|              
54*|1 + --------------|*cos(-1 + 3*x)
   |       2          |              
   \    sin (-1 + 3*x)/              
-------------------------------------
            sin(-1 + 3*x)            
$$\frac{54 \left(1 + \frac{\cos^{2}{\left(3 x - 1 \right)}}{\sin^{2}{\left(3 x - 1 \right)}}\right) \cos{\left(3 x - 1 \right)}}{\sin{\left(3 x - 1 \right)}}$$
The graph
Derivative of y=ln(sin(3x-1))