Question

write the function in the form y = f(u) and u = g(x). then find $\frac{dy}{dx}$ as a function of x.
$y=left(-1 - \frac{2x}{3}
ight)^{-3}$
u =
(type an expression using x as the variable.)

Explanation:

Step1: Define u and y

Let $u=-1 - \frac{2x}{3}$, then $y = u^{-3}$.

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

Using the power - rule for differentiation, if $y = u^{-3}$, then $\frac{dy}{du}=-3u^{-4}$.

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

Since $u=-1-\frac{2x}{3}$, then $\frac{du}{dx}=-\frac{2}{3}$.

Step4: Use the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=-3u^{-4}$ and $\frac{du}{dx}=-\frac{2}{3}$ into the chain - rule formula. We get $\frac{dy}{dx}=(-3u^{-4})\cdot(-\frac{2}{3}) = 2u^{-4}$.

Step5: Substitute u back in terms of x

Since $u=-1-\frac{2x}{3}$, then $\frac{dy}{dx}=2(-1 - \frac{2x}{3})^{-4}$.

Answer:

$u=-1-\frac{2x}{3}$, $\frac{dy}{dx}=2(-1 - \frac{2x}{3})^{-4}$