Derivative of e^xsinx-3e^xcosx

1. Differentiate $e^{x} \sin{\left(x \right)} - 3 e^{x} \cos{\left(x \right)}$ term by term:

   1. Apply the product rule:

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

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

      1. The derivative of $e^{x}$ is itself.

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

      1. The derivative of sine is cosine:

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

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

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

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

         1. Apply the product rule:

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

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

            1. The derivative of $e^{x}$ is itself.

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

            1. The derivative of cosine is negative sine:

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

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

         So, the result is: $- 3 e^{x} \sin{\left(x \right)} + 3 e^{x} \cos{\left(x \right)}$

      So, the result is: $3 e^{x} \sin{\left(x \right)} - 3 e^{x} \cos{\left(x \right)}$

   The result is: $4 e^{x} \sin{\left(x \right)} - 2 e^{x} \cos{\left(x \right)}$

2. Now simplify:

   $$2 \cdot \left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}$$

The answer is:

$$2 \cdot \left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}$$