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

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

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

      1. Apply the product rule:

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

         $f{\left(x \right)} = 5 x - 6$; 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 $-6$ 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 sine is cosine:

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

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

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

   2. The derivative of the constant $-8$ 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:

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