Question

1. find $\frac{dy}{dx}$ for the given pair of functions, $y = 5u^{2}+3u$, where $u=x^{3}+1$

Explanation:

Step1: Find $\frac{dy}{du}$

Differentiate $y = 5u^{2}+3u$ with respect to $u$. Using the power - rule $\frac{d}{du}(au^{n})=nau^{n - 1}$, we have $\frac{dy}{du}=\frac{d}{du}(5u^{2}+3u)=10u + 3$.

Step2: Find $\frac{du}{dx}$

Differentiate $u=x^{3}+1$ with respect to $x$. Using the power - rule, $\frac{du}{dx}=\frac{d}{dx}(x^{3}+1)=3x^{2}$.

Step3: Use the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=10u + 3$ and $\frac{du}{dx}=3x^{2}$ into the chain - rule formula. Then replace $u$ with $x^{3}+1$. So $\frac{dy}{dx}=(10(x^{3}+1)+3)\cdot3x^{2}=(10x^{3}+10 + 3)\cdot3x^{2}=(10x^{3}+13)\cdot3x^{2}=30x^{5}+39x^{2}$.

Answer:

$30x^{5}+39x^{2}$