Mister Exam

Derivative of 3sin(2x)−5cos(2x)

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

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Piecewise:

The solution

You have entered [src]
3*sin(2*x) - 5*cos(2*x)
3sin(2x)5cos(2x)3 \sin{\left(2 x \right)} - 5 \cos{\left(2 x \right)}
3*sin(2*x) - 5*cos(2*x)
Detail solution
  1. Differentiate 3sin(2x)5cos(2x)3 \sin{\left(2 x \right)} - 5 \cos{\left(2 x \right)} term by term:

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

      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)}

      So, the result is: 6cos(2x)6 \cos{\left(2 x \right)}

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

      1. Let u=2xu = 2 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 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:

        2sin(2x)- 2 \sin{\left(2 x \right)}

      So, the result is: 10sin(2x)10 \sin{\left(2 x \right)}

    The result is: 10sin(2x)+6cos(2x)10 \sin{\left(2 x \right)} + 6 \cos{\left(2 x \right)}


The answer is:

10sin(2x)+6cos(2x)10 \sin{\left(2 x \right)} + 6 \cos{\left(2 x \right)}

The graph
02468-8-6-4-2-1010-2525
The first derivative [src]
6*cos(2*x) + 10*sin(2*x)
10sin(2x)+6cos(2x)10 \sin{\left(2 x \right)} + 6 \cos{\left(2 x \right)}
The second derivative [src]
4*(-3*sin(2*x) + 5*cos(2*x))
4(3sin(2x)+5cos(2x))4 \left(- 3 \sin{\left(2 x \right)} + 5 \cos{\left(2 x \right)}\right)
3-я производная [src]
-8*(3*cos(2*x) + 5*sin(2*x))
8(5sin(2x)+3cos(2x))- 8 \left(5 \sin{\left(2 x \right)} + 3 \cos{\left(2 x \right)}\right)
The third derivative [src]
-8*(3*cos(2*x) + 5*sin(2*x))
8(5sin(2x)+3cos(2x))- 8 \left(5 \sin{\left(2 x \right)} + 3 \cos{\left(2 x \right)}\right)