Derivative of y=(3x+5)cos²x

1. Apply the product rule:

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

   $f{\left(x \right)} = 3 x + 5$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $3 x + 5$ 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: $3$

      2. The derivative of the constant $5$ is zero.

      The result is: $3$

   $g{\left(x \right)} = \cos^{2}{\left(x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = \cos{\left(x \right)}$.

   2. Apply the power rule: $u^{2}$ goes to $2 u$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$:

      1. The derivative of cosine is negative sine:

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

      The result of the chain rule is:

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

   The result is: $- 2 \cdot \left(3 x + 5\right) \sin{\left(x \right)} \cos{\left(x \right)} + 3 \cos^{2}{\left(x \right)}$

2. Now simplify:

   $$\left(\left(- 6 x - 10\right) \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) \cos{\left(x \right)}$$

The answer is:

$$\left(\left(- 6 x - 10\right) \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) \cos{\left(x \right)}$$