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e^(-x)*sin(2*x)

Derivative of e^(-x)*sin(2*x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 -x         
E  *sin(2*x)
$$e^{- x} \sin{\left(2 x \right)}$$
E^(-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. The derivative of is itself.

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

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