Derivative of 1/sin^4x

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     1   
1*-------
     4   
  sin (x)

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

d /     1   \
--|1*-------|
dx|     4   |
  \  sin (x)/

$$\frac{d}{d x} 1 \cdot \frac{1}{\sin^{4}{\left(x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 1$ and $g{\left(x \right)} = \sin^{4}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
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:
  
  $4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

$- \frac{4 \cos{\left(x \right)}}{\sin^{5}{\left(x \right)}}$

The answer is:

$- \frac{4 \cos{\left(x \right)}}{\sin^{5}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  -4*cos(x)   
--------------
          4   
sin(x)*sin (x)

$$- \frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)} \sin^{4}{\left(x \right)}}$$

Simplify

The second derivative [src]

  /         2   \
  |    5*cos (x)|
4*|1 + ---------|
  |        2    |
  \     sin (x) /
-----------------
        4        
     sin (x)

$$\frac{4 \cdot \left(1 + \frac{5 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{\sin^{4}{\left(x \right)}}$$

Simplify

The third derivative [src]

   /          2   \       
   |    15*cos (x)|       
-8*|7 + ----------|*cos(x)
   |        2     |       
   \     sin (x)  /       
--------------------------
            5             
         sin (x)

$$- \frac{8 \cdot \left(7 + \frac{15 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin^{5}{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of 1/sin^4x

The graph:

Piecewise:

The solution