Derivative of sin^-1(x)

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  1   
------
sin(x)

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

1/sin(x)

Detail solution

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]

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

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

Simplify

The second derivative [src]

         2   
    2*cos (x)
1 + ---------
        2    
     sin (x) 
-------------
    sin(x)

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

Simplify

The third derivative [src]

 /         2   \        
 |    6*cos (x)|        
-|5 + ---------|*cos(x) 
 |        2    |        
 \     sin (x) /        
------------------------
           2            
        sin (x)

$$- \frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

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Derivative of sin^-1(x)

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The solution