Mister Exam

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Derivative of:
Derivative of sin(pi/4)
Derivative of y=ln(sinx^2)
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Derivative of y=-x³(3x⁴-2)
Identical expressions
y=ln(sinx^ two)
y equally ln( sinus of x squared )
y equally ln( sinus of x to the power of two)
y=ln(sinx2)
y=lnsinx2
y=ln(sinx²)
y=ln(sinx to the power of 2)
y=lnsinx^2
Similar expressions
lnsin(x^2+1)

The derivative of the function/ y=ln(sinx^2)

Derivative of y=ln(sinx^2)

The solution

You have entered [src]

   /   2   \
log\sin (x)/

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

d /   /   2   \\
--\log\sin (x)//
dx

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

Transform

Detail solution

Let $u = \sin^{2}{\left(x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin^{2}{\left(x \right)}$ :
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:
  
  $2 \sin{\left(x \right)} \cos{\left(x \right)}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=ln(sinx^2)

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The solution