Derivative of (7x+6)*sin(5x+4)

1. Apply the product rule:

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

   $f{\left(x \right)} = 7 x + 6$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $7 x + 6$ 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: $7$

      2. The derivative of the constant $6$ is zero.

      The result is: $7$

   $g{\left(x \right)} = \sin{\left(5 x + 4 \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 5 x + 4$.

   2. The derivative of sine is cosine:

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x + 4\right)$:

      1. Differentiate $5 x + 4$ 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 $4$ is zero.

         The result is: $5$

      The result of the chain rule is:

      $$5 \cos{\left(5 x + 4 \right)}$$

   The result is: $5 \left(7 x + 6\right) \cos{\left(5 x + 4 \right)} + 7 \sin{\left(5 x + 4 \right)}$

2. Now simplify:

   $$\left(35 x + 30\right) \cos{\left(5 x + 4 \right)} + 7 \sin{\left(5 x + 4 \right)}$$

The answer is:

$$\left(35 x + 30\right) \cos{\left(5 x + 4 \right)} + 7 \sin{\left(5 x + 4 \right)}$$