Derivative of csc(x)

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csc(x)

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

csc(x)

Detail solution

Rewrite the function to be differentiated:

$\csc{\left(x \right)} = \frac{1}{\sin{\left(x \right)}}$
Let $u = \sin{\left(x \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
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 answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-cot(x)*csc(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

 /         2   \              
-\5 + 6*cot (x)/*cot(x)*csc(x)

$$- \left(6 \cot^{2}{\left(x \right)} + 5\right) \cot{\left(x \right)} \csc{\left(x \right)}$$

Simplify

The graph