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The derivative of the function/ -cos(sin(x))/sin(x)^2

Derivative of -cos(sin(x))/sin(x)^2

The solution

You have entered [src]

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

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

(-cos(sin(x)))/sin(x)^2

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = - \cos{\left(\sin{\left(x \right)} \right)}$ and $g{\left(x \right)} = \sin^{2}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \sin{\left(x \right)}$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  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:
    
    $- \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}$
  So, the result is: $\sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
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:
  
  $2 \sin{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{\sin^{2}{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{\sin^{4}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/                                                                                                                                             /         2   \            \       
|                                                                                                                                             |    3*cos (x)|            |       
|                                                                                                                                           8*|2 + ---------|*cos(sin(x))|       
|                                                              /                        2               \     /         2   \                 |        2    |            |       
|                  2                                         6*\sin(x)*sin(sin(x)) - cos (x)*cos(sin(x))/     |    3*cos (x)|                 \     sin (x) /            |       
|-sin(sin(x)) - cos (x)*sin(sin(x)) - 3*cos(sin(x))*sin(x) + -------------------------------------------- + 6*|1 + ---------|*sin(sin(x)) + -----------------------------|*cos(x)
|                                                                               sin(x)                        |        2    |                           sin(x)           |       
\                                                                                                             \     sin (x) /                                            /       
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                        2                                                                                        
                                                                                     sin (x)

$$\frac{\left(6 \left(1 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(\sin{\left(x \right)} \right)} + \frac{8 \left(2 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}} + \frac{6 \left(\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}\right)}{\sin{\left(x \right)}} - 3 \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)} - \sin{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$

Simplify

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Derivative of -cos(sin(x))/sin(x)^2

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The solution