Question

hw14 the chain rule (target c4; §3.6) due: thu oct 9, 2025 11:59pm attempt 1 in progress next up: submit assignment details no additional details were added for this assignment. hw14 the chain rule (target c4; §3.6) score: 4/11 answered: 4/11 question 5 if (f(x)=(3x + 2)^{-2}), find (f(x)) question help: video message instructor submit question

Explanation:

Step1: Identify outer - inner functions

Let $u = 3x + 2$, then $y = u^{-2}$.

Step2: Differentiate outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=-2u^{-3}$ using the power rule $\frac{d}{du}(u^n)=nu^{n - 1}$.

Step3: Differentiate inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=3$.

Step4: Apply chain rule

By the chain rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$: $\frac{dy}{dx}=-2u^{-3}\cdot3$.

Step5: Substitute $u$ back

Replace $u = 3x+2$ into the expression: $\frac{dy}{dx}=-6(3x + 2)^{-3}$.

Answer:

$-6(3x + 2)^{-3}$