Question

write the function in the form y = f(u) and u = g(x). then find $\frac{dy}{dx}$ as a function of x
$y=(5 - \frac{7x}{4})^{-4}$
$u = 5-\frac{7x}{4}$
(type an expression using x as the variable.)
$y = u^{-4}$
(type an expression using u as the variable.)
$\frac{dy}{dx}=square$
(type an expression using x as the variable.)

Explanation:

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

Differentiate $y = u^{-4}$ with respect to $u$ using the power - rule $\frac{d}{du}(u^n)=nu^{n - 1}$. So, $\frac{dy}{du}=-4u^{-5}$.

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

Differentiate $u = 5-\frac{7x}{4}$ with respect to $x$. The derivative of a constant is 0 and the derivative of $-\frac{7x}{4}$ is $-\frac{7}{4}$. So, $\frac{du}{dx}=-\frac{7}{4}$.

Step3: Use the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=-4u^{-5}$ and $\frac{du}{dx}=-\frac{7}{4}$ into the chain - rule formula. We get $\frac{dy}{dx}=(-4u^{-5})\cdot(-\frac{7}{4}) = 7u^{-5}$.

Step4: Substitute $u = 5-\frac{7x}{4}$ back in

Replace $u$ with $5-\frac{7x}{4}$ in the expression for $\frac{dy}{dx}$. So, $\frac{dy}{dx}=\frac{7}{(5-\frac{7x}{4})^{5}}$.

Answer:

$\frac{7}{(5 - \frac{7x}{4})^{5}}$