Derivative of (5x-6)cosx-5sinx-8

1. Differentiate $\left(5 x - 6\right) \cos{\left(x \right)} - 5 \sin{\left(x \right)} - 8$ term by term:

   1. Apply the product rule:

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

      $f{\left(x \right)} = 5 x - 6$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Differentiate $5 x - 6$ term by term:

         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: $5$

         2. The derivative of the constant $\left(-1\right) 6$ is zero.

         The result is: $5$

      $g{\left(x \right)} = \cos{\left(x \right)}$; 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: $- \left(5 x - 6\right) \sin{\left(x \right)} + 5 \cos{\left(x \right)}$

   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. The derivative of sine is cosine:

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

         So, the result is: $5 \cos{\left(x \right)}$

      So, the result is: $- 5 \cos{\left(x \right)}$

   3. The derivative of the constant $\left(-1\right) 8$ is zero.

   The result is: $- \left(5 x - 6\right) \sin{\left(x \right)}$

2. Now simplify:

   $$\left(6 - 5 x\right) \sin{\left(x \right)}$$

The answer is:

$$\left(6 - 5 x\right) \sin{\left(x \right)}$$