Mister Exam

Derivative of e^(2x)cosx

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 2*x       
E   *cos(x)
$$e^{2 x} \cos{\left(x \right)}$$
E^(2*x)*cos(x)
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. The derivative of cosine is negative sine:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
   2*x                    2*x
- e   *sin(x) + 2*cos(x)*e   
$$- e^{2 x} \sin{\left(x \right)} + 2 e^{2 x} \cos{\left(x \right)}$$
The second derivative [src]
                        2*x
(-4*sin(x) + 3*cos(x))*e   
$$\left(- 4 \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) e^{2 x}$$
The third derivative [src]
                         2*x
(-11*sin(x) + 2*cos(x))*e   
$$\left(- 11 \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{2 x}$$