Derivative of (sin(5x+3))^5

1. Let $u = \sin{\left(5 x + 3 \right)}$.

2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$

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

   1. Let $u = 5 x + 3$.

   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 $\frac{d}{d x} \left(5 x + 3\right)$:

      1. Differentiate $5 x + 3$ 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 $3$ is zero.

         The result is: $5$

      The result of the chain rule is:

      $$5 \cos{\left(5 x + 3 \right)}$$

   The result of the chain rule is:

   $$25 \sin^{4}{\left(5 x + 3 \right)} \cos{\left(5 x + 3 \right)}$$

4. Now simplify:

   $$25 \sin^{4}{\left(5 x + 3 \right)} \cos{\left(5 x + 3 \right)}$$

The answer is:

$$25 \sin^{4}{\left(5 x + 3 \right)} \cos{\left(5 x + 3 \right)}$$