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20/3*sin(x)*cos(x)
The derivative of the function/ 20/3*sin(x)*cos(x)

Derivative of 20/3*sin(x)*cos(x)

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

You have entered [src] 
20*sin(x)*cos(x)
----------------
       3
20sin⁡(x)cos⁡(x)3\frac{20 \sin{\left(x \right)} \cos{\left(x \right)}}{3} 
d /20*sin(x)*cos(x)\
--|----------------|
dx\       3        /
ddx20sin⁡(x)cos⁡(x)3\frac{d}{d x} \frac{20 \sin{\left(x \right)} \cos{\left(x \right)}}{3}
Transform
Detail solution

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

    1. Apply the product rule:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

      f(x)=cos⁡(x)f{\left(x \right)} = \cos{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. The derivative of cosine is negative sine:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

      g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\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)}

      The result is: −sin⁡2(x)+cos⁡2(x)- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}

    So, the result is: −20sin⁡2(x)3+20cos⁡2(x)3- \frac{20 \sin^{2}{\left(x \right)}}{3} + \frac{20 \cos^{2}{\left(x \right)}}{3}

  2. Now simplify:

    20cos⁡(2x)3\frac{20 \cos{\left(2 x \right)}}{3}

The answer is:

20cos⁡(2x)3\frac{20 \cos{\left(2 x \right)}}{3}

The graph
02468-8-6-4-2-1010-1010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        2            2   
  20*sin (x)   20*cos (x)
- ---------- + ----------
      3            3
−20sin⁡2(x)3+20cos⁡2(x)3- \frac{20 \sin^{2}{\left(x \right)}}{3} + \frac{20 \cos^{2}{\left(x \right)}}{3}
Simplify
The second derivative [src] 
-80*cos(x)*sin(x)
-----------------
        3
−80sin⁡(x)cos⁡(x)3- \frac{80 \sin{\left(x \right)} \cos{\left(x \right)}}{3}
Simplify
The third derivative [src] 
   /   2         2   \
80*\sin (x) - cos (x)/
----------------------
          3
80(sin⁡2(x)−cos⁡2(x))3\frac{80 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{3}
Simplify
The graph
Derivative of 20/3*sin(x)*cos(x)
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