Question

write the function below in the form y = f(u) and u = g(x). then find $\frac{dy}{dx}$ as a function of x

$y=sqrt{3x^{2}+3x + 5}$

write the function in the form y = f(u) and u = g(x). choose the correct answer below

a. $y=sqrt{u}$ and $u = 3x^{2}+3x + 5$
b. $y = 3u^{2}+3u + 5$ and $u = x$
c. $y = u$ and $u = 3x^{2}+3x + 5$
d. $y = 3u^{2}+3u + 5$ and $u=sqrt{x}$

Explanation:

Step1: Identify f(u) and g(x)

We rewrite $y = \sqrt{3x^{2}+3x + 5}$ as a composition. Let $u=3x^{2}+3x + 5$ and $y = \sqrt{u}$. So the correct form is $y = f(u)=\sqrt{u}$ and $u = g(x)=3x^{2}+3x + 5$, which is option A.

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

If $y=\sqrt{u}=u^{\frac{1}{2}}$, then by the power - rule $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}=\frac{1}{2\sqrt{u}}$.

Step3: Find $\frac{du}{dx}$

If $u = 3x^{2}+3x + 5$, then $\frac{du}{dx}=6x + 3$.

Step4: Use the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\frac{1}{2\sqrt{u}}$ and $\frac{du}{dx}=6x + 3$ into the chain - rule formula. Replace $u$ with $3x^{2}+3x + 5$. So $\frac{dy}{dx}=\frac{6x + 3}{2\sqrt{3x^{2}+3x + 5}}$.

Answer:

A. $y=\sqrt{u}$ and $u = 3x^{2}+3x + 5$
$\frac{dy}{dx}=\frac{6x + 3}{2\sqrt{3x^{2}+3x + 5}}$