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

1. Let $u = 3 x^{3} + 5 x$.

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

   1. Differentiate $3 x^{3} + 5 x$ 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 a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

         So, the result is: $9 x^{2}$

      The result is: $9 x^{2} + 5$

   The result of the chain rule is:

   $$\left(9 x^{2} + 5\right) \cos{\left(3 x^{3} + 5 x \right)}$$

4. Now simplify:

   $$\left(9 x^{2} + 5\right) \cos{\left(x \left(3 x^{2} + 5\right) \right)}$$

The answer is: