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Derivative of e^(x*(-2))*sin(3*x)

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

The graph:

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The solution

You have entered [src]
 x*(-2)         
E      *sin(3*x)
$$e^{\left(-2\right) x} \sin{\left(3 x \right)}$$
E^(x*(-2))*sin(3*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. 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]
     x*(-2)                        x*(-2)
- 2*e      *sin(3*x) + 3*cos(3*x)*e      
$$- 2 e^{\left(-2\right) x} \sin{\left(3 x \right)} + 3 e^{\left(-2\right) x} \cos{\left(3 x \right)}$$
The second derivative [src]
                             -2*x
(-12*cos(3*x) - 5*sin(3*x))*e    
$$\left(- 5 \sin{\left(3 x \right)} - 12 \cos{\left(3 x \right)}\right) e^{- 2 x}$$
The third derivative [src]
                            -2*x
(9*cos(3*x) + 46*sin(3*x))*e    
$$\left(46 \sin{\left(3 x \right)} + 9 \cos{\left(3 x \right)}\right) e^{- 2 x}$$