Mister Exam

Derivative of e^(2x)sin2x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 2*x         
e   *sin(2*x)
$$e^{2 x} \sin{\left(2 x \right)}$$
d / 2*x         \
--\e   *sin(2*x)/
dx               
$$\frac{d}{d x} e^{2 x} \sin{\left(2 x \right)}$$
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Let .

    2. The derivative of is itself.

    3. Then, apply the chain rule. Multiply by :

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

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    ; to find :

    1. Let .

    2. The derivative of sine is cosine:

    3. Then, apply the chain rule. Multiply by :

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

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
            2*x      2*x         
2*cos(2*x)*e    + 2*e   *sin(2*x)
$$2 e^{2 x} \sin{\left(2 x \right)} + 2 e^{2 x} \cos{\left(2 x \right)}$$
The second derivative [src]
            2*x
8*cos(2*x)*e   
$$8 e^{2 x} \cos{\left(2 x \right)}$$
The third derivative [src]
                           2*x
16*(-sin(2*x) + cos(2*x))*e   
$$16 \left(- \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{2 x}$$
The graph
Derivative of e^(2x)sin2x