Derivative of y=ln(3sinx)

You have entered [src]

log(3*sin(x))

$$\log{\left(3 \sin{\left(x \right)} \right)}$$

d                
--(log(3*sin(x)))
dx

$$\frac{d}{d x} \log{\left(3 \sin{\left(x \right)} \right)}$$

Transform

Detail solution

Let $u = 3 \sin{\left(x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 \sin{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  So, the result is: $3 \cos{\left(x \right)}$
The result of the chain rule is:

$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

$\frac{1}{\tan{\left(x \right)}}$

The answer is:

$\frac{1}{\tan{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(x)
------
sin(x)

$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$

Simplify

The second derivative [src]

 /       2   \
 |    cos (x)|
-|1 + -------|
 |       2   |
 \    sin (x)/

$$- (1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}})$$

Simplify

The third derivative [src]

  /       2   \       
  |    cos (x)|       
2*|1 + -------|*cos(x)
  |       2   |       
  \    sin (x)/       
----------------------
        sin(x)

$$\frac{2 \cdot \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=ln(3sinx)

The graph:

Piecewise:

The solution