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The derivative of the function/ sin(2*x)/((5*x))

Derivative of sin(2*x)/((5*x))

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

You have entered [src] 
sin(2*x)
--------
  5*x
sin⁡(2x)5x\frac{\sin{\left(2 x \right)}}{5 x} 
sin(2*x)/((5*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)=sin⁡(2x)f{\left(x \right)} = \sin{\left(2 x \right)} and g(x)=5xg{\left(x \right)} = 5 x.

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

    1. Let u=2xu = 2 x.

    2. The derivative of sine is cosine:

      ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx2x\frac{d}{d x} 2 x:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: xx goes to 11

        So, the result is: 22

      The result of the chain rule is:

      2cos⁡(2x)2 \cos{\left(2 x \right)}

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

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: xx goes to 11

      So, the result is: 55

    Now plug in to the quotient rule:

    10xcos⁡(2x)−5sin⁡(2x)25x2\frac{10 x \cos{\left(2 x \right)} - 5 \sin{\left(2 x \right)}}{25 x^{2}}

  2. Now simplify:

    2xcos⁡(2x)−sin⁡(2x)5x2\frac{2 x \cos{\left(2 x \right)} - \sin{\left(2 x \right)}}{5 x^{2}}

The answer is:

2xcos⁡(2x)−sin⁡(2x)5x2\frac{2 x \cos{\left(2 x \right)} - \sin{\left(2 x \right)}}{5 x^{2}}

The graph
02468-8-6-4-2-10101.0-1.0
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   1             sin(2*x)
2*---*cos(2*x) - --------
  5*x                 2  
                   5*x
215xcos⁡(2x)−sin⁡(2x)5x22 \frac{1}{5 x} \cos{\left(2 x \right)} - \frac{\sin{\left(2 x \right)}}{5 x^{2}}
Simplify
The second derivative [src] 
  /              sin(2*x)   2*cos(2*x)\
2*|-2*sin(2*x) + -------- - ----------|
  |                  2          x     |
  \                 x                 /
---------------------------------------
                  5*x
2(−2sin⁡(2x)−2cos⁡(2x)x+sin⁡(2x)x2)5x\frac{2 \left(- 2 \sin{\left(2 x \right)} - \frac{2 \cos{\left(2 x \right)}}{x} + \frac{\sin{\left(2 x \right)}}{x^{2}}\right)}{5 x}
Simplify
The third derivative [src] 
  /              3*sin(2*x)   6*sin(2*x)   6*cos(2*x)\
2*|-4*cos(2*x) - ---------- + ---------- + ----------|
  |                   3           x             2    |
  \                  x                         x     /
------------------------------------------------------
                         5*x
2(−4cos⁡(2x)+6sin⁡(2x)x+6cos⁡(2x)x2−3sin⁡(2x)x3)5x\frac{2 \left(- 4 \cos{\left(2 x \right)} + \frac{6 \sin{\left(2 x \right)}}{x} + \frac{6 \cos{\left(2 x \right)}}{x^{2}} - \frac{3 \sin{\left(2 x \right)}}{x^{3}}\right)}{5 x}
Simplify
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