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

Derivative of cos(x)*sin(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
cos(x)*sin(x)
sin⁡(x)cos⁡(x)\sin{\left(x \right)} \cos{\left(x \right)} 
cos(x)*sin(x)
Detail solution

  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)}

  2. Now simplify:

    cos⁡(2x)\cos{\left(2 x \right)}

The answer is:

cos⁡(2x)\cos{\left(2 x \right)}

The graph
02468-8-6-4-2-10102-2
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   2         2   
cos (x) - sin (x)
−sin⁡2(x)+cos⁡2(x)- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}
Simplify
The second derivative [src] 
-4*cos(x)*sin(x)
−4sin⁡(x)cos⁡(x)- 4 \sin{\left(x \right)} \cos{\left(x \right)}
Simplify
The third derivative [src] 
  /   2         2   \
4*\sin (x) - cos (x)/
4(sin⁡2(x)−cos⁡2(x))4 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)
Simplify
The graph
Derivative of cos(x)*sin(x)
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