Mister Exam

Derivative of 10cos5x*sin5x

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

You have entered [src]
10*cos(5*x)*sin(5*x)
sin(5x)10cos(5x)\sin{\left(5 x \right)} 10 \cos{\left(5 x \right)}
(10*cos(5*x))*sin(5*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)=10cos(5x)f{\left(x \right)} = 10 \cos{\left(5 x \right)}; 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. Let u=5xu = 5 x.

      2. The derivative of cosine is negative sine:

        dducos(u)=sin(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddx5x\frac{d}{d x} 5 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: 55

        The result of the chain rule is:

        5sin(5x)- 5 \sin{\left(5 x \right)}

      So, the result is: 50sin(5x)- 50 \sin{\left(5 x \right)}

    g(x)=sin(5x)g{\left(x \right)} = \sin{\left(5 x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=5xu = 5 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 ddx5x\frac{d}{d x} 5 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: 55

      The result of the chain rule is:

      5cos(5x)5 \cos{\left(5 x \right)}

    The result is: 50sin2(5x)+50cos2(5x)- 50 \sin^{2}{\left(5 x \right)} + 50 \cos^{2}{\left(5 x \right)}

  2. Now simplify:

    50cos(10x)50 \cos{\left(10 x \right)}


The answer is:

50cos(10x)50 \cos{\left(10 x \right)}

The graph
02468-8-6-4-2-1010-100100
The first derivative [src]
        2              2     
- 50*sin (5*x) + 50*cos (5*x)
50sin2(5x)+50cos2(5x)- 50 \sin^{2}{\left(5 x \right)} + 50 \cos^{2}{\left(5 x \right)}
The second derivative [src]
-1000*cos(5*x)*sin(5*x)
1000sin(5x)cos(5x)- 1000 \sin{\left(5 x \right)} \cos{\left(5 x \right)}
The third derivative [src]
     /   2           2     \
5000*\sin (5*x) - cos (5*x)/
5000(sin2(5x)cos2(5x))5000 \left(\sin^{2}{\left(5 x \right)} - \cos^{2}{\left(5 x \right)}\right)