Derivative of y=sin^-1(3x)

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   1    
--------
sin(3*x)

$$\frac{1}{\sin{\left(3 x \right)}}$$

1/sin(3*x)

Detail solution

Let $u = \sin{\left(3 x \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x \right)}$ :
1. Let $u = 3 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $3$
  The result of the chain rule is:
  
  $3 \cos{\left(3 x \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-3*cos(3*x)
-----------
    2      
 sin (3*x)

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

Simplify

The second derivative [src]

  /         2     \
  |    2*cos (3*x)|
9*|1 + -----------|
  |        2      |
  \     sin (3*x) /
-------------------
      sin(3*x)

$$\frac{9 \left(1 + \frac{2 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right)}{\sin{\left(3 x \right)}}$$

Simplify

The third derivative [src]

    /         2     \         
    |    6*cos (3*x)|         
-27*|5 + -----------|*cos(3*x)
    |        2      |         
    \     sin (3*x) /         
------------------------------
             2                
          sin (3*x)

$$- \frac{27 \left(5 + \frac{6 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right) \cos{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}$$

Simplify

The graph

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Derivative of y=sin^-1(3x)

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The solution