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The derivative of the function/ y=-4/x^5(-sinx)

Derivative of y=-4/x^5(-sinx)

Function f() - derivative -N order at the point
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The graph:

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The solution

You have entered [src] 
-4           
---*(-sin(x))
  5          
 x
−4x5(−sin⁡(x))- \frac{4}{x^{5}} \left(- \sin{\left(x \right)}\right) 
(-4/x^5)*(-sin(x))
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)=4sin⁡(x)f{\left(x \right)} = 4 \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 a constant times a function is the constant times the derivative of the function.

      1. The derivative of sine is cosine:

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

      So, the result is: 4cos⁡(x)4 \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:

    4x5cos⁡(x)−20x4sin⁡(x)x10\frac{4 x^{5} \cos{\left(x \right)} - 20 x^{4} \sin{\left(x \right)}}{x^{10}}

  2. Now simplify:

    4(xcos⁡(x)−5sin⁡(x))x6\frac{4 \left(x \cos{\left(x \right)} - 5 \sin{\left(x \right)}\right)}{x^{6}}

The answer is:

4(xcos⁡(x)−5sin⁡(x))x6\frac{4 \left(x \cos{\left(x \right)} - 5 \sin{\left(x \right)}\right)}{x^{6}}

The graph
02468-8-6-4-2-1010-50000005000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  20*sin(x)   4*cos(x)
- --------- + --------
       6          5   
      x          x
4cos⁡(x)x5−20sin⁡(x)x6\frac{4 \cos{\left(x \right)}}{x^{5}} - \frac{20 \sin{\left(x \right)}}{x^{6}}
Simplify
The second derivative [src] 
  /          10*cos(x)   30*sin(x)\
4*|-sin(x) - --------- + ---------|
  |              x            2   |
  \                          x    /
-----------------------------------
                  5                
                 x
4(−sin⁡(x)−10cos⁡(x)x+30sin⁡(x)x2)x5\frac{4 \left(- \sin{\left(x \right)} - \frac{10 \cos{\left(x \right)}}{x} + \frac{30 \sin{\left(x \right)}}{x^{2}}\right)}{x^{5}}
Simplify
The third derivative [src] 
  /          210*sin(x)   15*sin(x)   90*cos(x)\
4*|-cos(x) - ---------- + --------- + ---------|
  |               3           x            2   |
  \              x                        x    /
------------------------------------------------
                        5                       
                       x
4(−cos⁡(x)+15sin⁡(x)x+90cos⁡(x)x2−210sin⁡(x)x3)x5\frac{4 \left(- \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}}\right)}{x^{5}}
Simplify
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