Derivative of y=sinx*(8e^x-6x)

1. Apply the product rule:

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

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

   1. The derivative of sine is cosine:

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

   $g{\left(x \right)} = - 6 x + 8 e^{x}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $- 6 x + 8 e^{x}$ term by term:

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

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

         So, the result is: $8 e^{x}$

      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 power rule: $x$ goes to $1$

            So, the result is: $6$

         So, the result is: $-6$

      The result is: $8 e^{x} - 6$

   The result is: $\left(- 6 x + 8 e^{x}\right) \cos{\left(x \right)} + \left(8 e^{x} - 6\right) \sin{\left(x \right)}$

2. Now simplify:

   $$2 \left(- 3 x + 4 e^{x}\right) \cos{\left(x \right)} + 2 \cdot \left(4 e^{x} - 3\right) \sin{\left(x \right)}$$

The answer is:

$$2 \left(- 3 x + 4 e^{x}\right) \cos{\left(x \right)} + 2 \cdot \left(4 e^{x} - 3\right) \sin{\left(x \right)}$$