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

Derivative of y=sinx/x^5

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
sin(x)
------
   5  
  x
sin⁡(x)x5\frac{\sin{\left(x \right)}}{x^{5}} 
sin(x)/x^5
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=x5g{\left(x \right)} = x^{5}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Apply the power rule: x5x^{5} goes to 5x45 x^{4}

    Now plug in to the quotient rule:

    x5cos⁡(x)−5x4sin⁡(x)x10\frac{x^{5} \cos{\left(x \right)} - 5 x^{4} \sin{\left(x \right)}}{x^{10}}

  2. Now simplify:

    xcos⁡(x)−5sin⁡(x)x6\frac{x \cos{\left(x \right)} - 5 \sin{\left(x \right)}}{x^{6}}

The answer is:

xcos⁡(x)−5sin⁡(x)x6\frac{x \cos{\left(x \right)} - 5 \sin{\left(x \right)}}{x^{6}}

The graph
02468-8-6-4-2-1010-10000001000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
cos(x)   5*sin(x)
------ - --------
   5         6   
  x         x
cos⁡(x)x5−5sin⁡(x)x6\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
−sin⁡(x)−10cos⁡(x)x+30sin⁡(x)x2x5\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
−cos⁡(x)+15sin⁡(x)x+90cos⁡(x)x2−210sin⁡(x)x3x5\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
Derivative of y=sinx/x^5
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