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The derivative of the function/ csc^2

Derivative of csc^2

The solution

You have entered [src]

   2   
csc (x)

$$\csc^{2}{\left(x \right)}$$

csc(x)^2

Detail solution

Let $u = \csc{\left(x \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \csc{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\csc{\left(x \right)} = \frac{1}{\sin{\left(x \right)}}$
2. Let $u = \sin{\left(x \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. 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:
  
  $- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      2          
-2*csc (x)*cot(x)

$$- 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)}$$

Simplify

The second derivative [src]

     2    /         2   \
2*csc (x)*\1 + 3*cot (x)/

$$2 \left(3 \cot^{2}{\left(x \right)} + 1\right) \csc^{2}{\left(x \right)}$$

Simplify

The third derivative [src]

      2    /         2   \       
-8*csc (x)*\2 + 3*cot (x)/*cot(x)

$$- 8 \left(3 \cot^{2}{\left(x \right)} + 2\right) \cot{\left(x \right)} \csc^{2}{\left(x \right)}$$

Simplify

The graph

Derivative of csc^2