Question

differentiate the function.
( y = (7x + 8)^5 )

( \frac{dy}{dx} = square )

Explanation:

Step1: Apply the chain rule

The chain rule states that if \( y = u^n \) and \( u = f(x) \), then \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \). Let \( u = 7x + 8 \), so \( y=u^{5} \).
First, find \( \frac{dy}{du} \): for \( y = u^{5} \), using the power rule \( \frac{d}{du}(u^{n})=nu^{n - 1} \), we get \( \frac{dy}{du}=5u^{4} \).

Step2: Find \( \frac{du}{dx} \)

For \( u = 7x+8 \), the derivative \( \frac{du}{dx}=\frac{d}{dx}(7x + 8)=7 \).

Step3: Multiply the two derivatives

Using the chain rule \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \), substitute \( \frac{dy}{du}=5u^{4} \) and \( \frac{du}{dx} = 7 \), and then replace \( u \) with \( 7x + 8 \).
So \( \frac{dy}{dx}=5(7x + 8)^{4}\cdot7=35(7x + 8)^{4} \).

Answer:

\( 35(7x + 8)^{4} \)