Derivative of ln(sint²)

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   /   2   \
log\sin (t)/

$$\log{\left(\sin^{2}{\left(t \right)} \right)}$$

log(sin(t)^2)

Detail solution

Let $u = \sin^{2}{\left(t \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d t} \sin^{2}{\left(t \right)}$ :
1. Let $u = \sin{\left(t \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \sin{\left(t \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$
  The result of the chain rule is:
  
  $2 \sin{\left(t \right)} \cos{\left(t \right)}$
The result of the chain rule is:

$\frac{2 \cos{\left(t \right)}}{\sin{\left(t \right)}}$
Now simplify:

$\frac{2}{\tan{\left(t \right)}}$

The answer is:

$\frac{2}{\tan{\left(t \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

2*cos(t)
--------
 sin(t)

$$\frac{2 \cos{\left(t \right)}}{\sin{\left(t \right)}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       2   \       
  |    cos (t)|       
4*|1 + -------|*cos(t)
  |       2   |       
  \    sin (t)/       
----------------------
        sin(t)

$$\frac{4 \left(1 + \frac{\cos^{2}{\left(t \right)}}{\sin^{2}{\left(t \right)}}\right) \cos{\left(t \right)}}{\sin{\left(t \right)}}$$

Simplify

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Identical expressions

Derivative of ln(sint²)

The graph:

Piecewise:

The solution