Derivative of y=lnsinx-½sin²x

You have entered [src]

                 2   
              sin (x)
log(sin(x)) - -------
                 2

$$\log{\left(\sin{\left(x \right)} \right)} - \frac{\sin^{2}{\left(x \right)}}{2}$$

log(sin(x)) - sin(x)^2/2

Detail solution

Differentiate $\log{\left(\sin{\left(x \right)} \right)} - \frac{\sin^{2}{\left(x \right)}}{2}$ term by term:
1. Let $u = \sin{\left(x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  So, the result is: $- \sin{\left(x \right)} \cos{\left(x \right)}$
The result is: $- \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

$\frac{\cos^{3}{\left(x \right)}}{\sin{\left(x \right)}}$

The answer is:

$\frac{\cos^{3}{\left(x \right)}}{\sin{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                            2   
        2         2      cos (x)
-1 + sin (x) - cos (x) - -------
                            2   
                         sin (x)

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

Simplify

The third derivative [src]

  /                       2   \       
  |  1                 cos (x)|       
2*|------ + 2*sin(x) + -------|*cos(x)
  |sin(x)                 3   |       
  \                    sin (x)/

$$2 \left(2 \sin{\left(x \right)} + \frac{1}{\sin{\left(x \right)}} + \frac{\cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right) \cos{\left(x \right)}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=lnsinx-½sin²x

The graph:

Piecewise:

The solution