Derivative of y=sin5x^-3

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    1    
---------
   3     
sin (5*x)

$$\frac{1}{\sin^{3}{\left(5 x \right)}}$$

sin(5*x)^(-3)

Detail solution

Let $u = \sin{\left(5 x \right)}$ .
Apply the power rule: $\frac{1}{u^{3}}$ goes to $- \frac{3}{u^{4}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(5 x \right)}$ :
1. Let $u = 5 x$ .
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 $\frac{d}{d x} 5 x$ :
  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: $5$
  The result of the chain rule is:
  
  $5 \cos{\left(5 x \right)}$
The result of the chain rule is:

$- \frac{15 \cos{\left(5 x \right)}}{\sin^{4}{\left(5 x \right)}}$

The answer is:

$- \frac{15 \cos{\left(5 x \right)}}{\sin^{4}{\left(5 x \right)}}$

The graph

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The first derivative [src]

-15*cos(5*x)
------------
    4       
 sin (5*x)

$$- \frac{15 \cos{\left(5 x \right)}}{\sin^{4}{\left(5 x \right)}}$$

Simplify

The second derivative [src]

   /         2     \
   |    4*cos (5*x)|
75*|1 + -----------|
   |        2      |
   \     sin (5*x) /
--------------------
        3           
     sin (5*x)

$$\frac{75 \left(1 + \frac{4 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right)}{\sin^{3}{\left(5 x \right)}}$$

Simplify

The third derivative [src]

     /           2     \         
     |     20*cos (5*x)|         
-375*|11 + ------------|*cos(5*x)
     |         2       |         
     \      sin (5*x)  /         
---------------------------------
               4                 
            sin (5*x)

$$- \frac{375 \left(11 + \frac{20 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \cos{\left(5 x \right)}}{\sin^{4}{\left(5 x \right)}}$$

Simplify

The graph

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Derivative of y=sin5x^-3

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The solution