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Derivative of y=sinx/x^5
Identical expressions
y=sinx/x^ five
y equally sinus of x divide by x to the power of 5
y equally sinus of x divide by x to the power of five
y=sinx/x5
y=sinx/x⁵
y=sinx divide by x^5

The derivative of the function/ y=sinx/x^5

Derivative of y=sinx/x^5

The solution

You have entered [src]

sin(x)
------
   5  
  x

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

sin(x)/x^5

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 $g{\left(x \right)} = x^{5}$ .

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: $x^{5}$ goes to $5 x^{4}$
Now plug in to the quotient rule:

$\frac{x^{5} \cos{\left(x \right)} - 5 x^{4} \sin{\left(x \right)}}{x^{10}}$
Now simplify:

$\frac{x \cos{\left(x \right)} - 5 \sin{\left(x \right)}}{x^{6}}$

The answer is:

$\frac{x \cos{\left(x \right)} - 5 \sin{\left(x \right)}}{x^{6}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(x)   5*sin(x)
------ - --------
   5         6   
  x         x

$$\frac{\cos{\left(x \right)}}{x^{5}} - \frac{5 \sin{\left(x \right)}}{x^{6}}$$

Simplify

The second derivative [src]

          10*cos(x)   30*sin(x)
-sin(x) - --------- + ---------
              x            2   
                          x    
-------------------------------
                5              
               x

$$\frac{- \sin{\left(x \right)} - \frac{10 \cos{\left(x \right)}}{x} + \frac{30 \sin{\left(x \right)}}{x^{2}}}{x^{5}}$$

Simplify

The third derivative [src]

          210*sin(x)   15*sin(x)   90*cos(x)
-cos(x) - ---------- + --------- + ---------
               3           x            2   
              x                        x    
--------------------------------------------
                      5                     
                     x

$$\frac{- \cos{\left(x \right)} + \frac{15 \sin{\left(x \right)}}{x} + \frac{90 \cos{\left(x \right)}}{x^{2}} - \frac{210 \sin{\left(x \right)}}{x^{3}}}{x^{5}}$$

Simplify

The graph

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Identical expressions

Derivative of y=sinx/x^5

The graph:

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The solution