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one /(sin^35x)
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one divide by ( sinus of cubed 5x)
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The derivative of the function/ 1/(sin^35x)

Derivative of 1/(sin^35x)

The solution

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     1    
1*--------
     35   
  sin  (x)

$$1 \cdot \frac{1}{\sin^{35}{\left(x \right)}}$$

d /     1    \
--|1*--------|
dx|     35   |
  \  sin  (x)/

$$\frac{d}{d x} 1 \cdot \frac{1}{\sin^{35}{\left(x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 1$ and $g{\left(x \right)} = \sin^{35}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{35}$ goes to $35 u^{34}$
3. 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:
  
  $35 \sin^{34}{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

$- \frac{35 \cos{\left(x \right)}}{\sin^{36}{\left(x \right)}}$

The answer is:

$- \frac{35 \cos{\left(x \right)}}{\sin^{36}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -35*cos(x)  
---------------
          35   
sin(x)*sin  (x)

$$- \frac{35 \cos{\left(x \right)}}{\sin{\left(x \right)} \sin^{35}{\left(x \right)}}$$

Simplify

The second derivative [src]

   /          2   \
   |    36*cos (x)|
35*|1 + ----------|
   |        2     |
   \     sin (x)  /
-------------------
         35        
      sin  (x)

$$\frac{35 \cdot \left(1 + \frac{36 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{\sin^{35}{\left(x \right)}}$$

Simplify

The third derivative [src]

    /              2   \       
    |      1332*cos (x)|       
-35*|107 + ------------|*cos(x)
    |           2      |       
    \        sin (x)   /       
-------------------------------
               36              
            sin  (x)

$$- \frac{35 \cdot \left(107 + \frac{1332 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin^{36}{\left(x \right)}}$$

Simplify

The graph

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Derivative of 1/(sin^35x)

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The solution