Derivative of sinx(4x+5)

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:

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

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

   1. Differentiate $4 x + 5$ 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: $4$

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

      The result is: $4$

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

2. Now simplify:

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

The answer is:

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