Derivative of exp(-ax)*sin(bx)

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = e^{- a x}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = - a x$.

   2. The derivative of $e^{u}$ is itself.

   3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} - a x$:

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

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $- a$

      The result of the chain rule is:

      $$- a e^{- a x}$$

   $g{\left(x \right)} = \sin{\left(b x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = b x$.

   2. The derivative of sine is cosine:

      $$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$$

   3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} b x$:

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

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $b$

      The result of the chain rule is:

      $$b \cos{\left(b x \right)}$$

   The result is: $- a e^{- a x} \sin{\left(b x \right)} + b e^{- a x} \cos{\left(b x \right)}$

2. Now simplify:

   $$\left(- a \sin{\left(b x \right)} + b \cos{\left(b x \right)}\right) e^{- a x}$$

The answer is:

$$\left(- a \sin{\left(b x \right)} + b \cos{\left(b x \right)}\right) e^{- a x}$$