Derivative of sinx/cos^4x

The solution

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 sin(x)
-------
   4   
cos (x)

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

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos^{4}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $- 4 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{4 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)} + \cos^{5}{\left(x \right)}}{\cos^{8}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

               2   
 cos(x)   4*sin (x)
------- + ---------
   4          5    
cos (x)    cos (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                            /          2   \
                       2    |    15*sin (x)|
                  8*sin (x)*|7 + ----------|
           2                |        2     |
     48*sin (x)             \     cos (x)  /
11 + ---------- + --------------------------
         2                    2             
      cos (x)              cos (x)          
--------------------------------------------
                     3                      
                  cos (x)

$$\frac{\frac{8 \left(\frac{15 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 7\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{48 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 11}{\cos^{3}{\left(x \right)}}$$

Simplify

The graph

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Derivative of sinx/cos^4x

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Piecewise:

The solution