find the derivative of the funtion. (hint: you will need to multiply first!)

Explanation:

Step1: Multiply the terms

$$ \begin{align*} y&=(x^{5}+3)\cdot4x^{3}\ y&=4x^{5}\cdot x^{3}+3\cdot4x^{3}\ y&=4x^{8}+12x^{3} \end{align*} $$

Step2: Differentiate term - by - term

Use the power rule $\frac{d}{dx}(x^{n}) = nx^{n - 1}$. For the term $4x^{8}$: $\frac{d}{dx}(4x^{8})=4\times8x^{8 - 1}=32x^{7}$ For the term $12x^{3}$: $\frac{d}{dx}(12x^{3})=12\times3x^{3 - 1}=36x^{2}$

Answer:

$32x^{7}+36x^{2}$