Mister Exam

Derivative of exp(-6x)*sin(2x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 -6*x         
e    *sin(2*x)
$$e^{- 6 x} \sin{\left(2 x \right)}$$
exp(-6*x)*sin(2*x)
Detail solution
  1. Apply the quotient rule, which is:

    and .

    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:

    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:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
     -6*x                        -6*x
- 6*e    *sin(2*x) + 2*cos(2*x)*e    
$$- 6 e^{- 6 x} \sin{\left(2 x \right)} + 2 e^{- 6 x} \cos{\left(2 x \right)}$$
The second derivative [src]
                              -6*x
8*(-3*cos(2*x) + 4*sin(2*x))*e    
$$8 \left(4 \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)}\right) e^{- 6 x}$$
4-я производная [src]
                                -6*x
64*(-24*cos(2*x) + 7*sin(2*x))*e    
$$64 \left(7 \sin{\left(2 x \right)} - 24 \cos{\left(2 x \right)}\right) e^{- 6 x}$$
The third derivative [src]
                                -6*x
16*(-9*sin(2*x) + 13*cos(2*x))*e    
$$16 \left(- 9 \sin{\left(2 x \right)} + 13 \cos{\left(2 x \right)}\right) e^{- 6 x}$$