Derivative of sinx/x^4

You have entered [src]

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

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

sin(x)/x^4

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 $g{\left(x \right)} = x^{4}$ .

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. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

          8*cos(x)   20*sin(x)
-sin(x) - -------- + ---------
             x            2   
                         x    
------------------------------
               4              
              x

$$\frac{- \sin{\left(x \right)} - \frac{8 \cos{\left(x \right)}}{x} + \frac{20 \sin{\left(x \right)}}{x^{2}}}{x^{4}}$$

Simplify

The third derivative [src]

          120*sin(x)   12*sin(x)   60*cos(x)
-cos(x) - ---------- + --------- + ---------
               3           x            2   
              x                        x    
--------------------------------------------
                      4                     
                     x

$$\frac{- \cos{\left(x \right)} + \frac{12 \sin{\left(x \right)}}{x} + \frac{60 \cos{\left(x \right)}}{x^{2}} - \frac{120 \sin{\left(x \right)}}{x^{3}}}{x^{4}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sinx/x^4

The graph:

Piecewise:

The solution