Question

differentiate the function.
( y = (11 - x)^{77} )

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

Explanation:

Step1: Apply the chain rule

Let \( u = 11 - x \), so \( y = u^{77} \). The chain rule states \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \).
First, find \( \frac{dy}{du} \): for \( y = u^{77} \), using the power rule \( \frac{d}{du}(u^n)=nu^{n - 1} \), we get \( \frac{dy}{du}=77u^{76} \).

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

For \( u = 11 - x \), the derivative \( \frac{du}{dx}=\frac{d}{dx}(11 - x)=- 1 \).

Step3: Combine using chain rule

Substitute \( u = 11 - x \) and the derivatives into the chain rule:
\( \frac{dy}{dx}=77(11 - x)^{76}\cdot(-1)=- 77(11 - x)^{76} \)

Answer:

\( -77(11 - x)^{76} \)