Question

differentiate the function.
( y = (5x + 14)^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 \( \frac{dy}{dx} = n u^{n - 1} \cdot \frac{du}{dx} \). Let \( u = 5x + 14 \), so \( y = u^5 \).
First, find the derivative of \( y \) with respect to \( u \): \( \frac{dy}{du} = 5u^{4} \).

Step2: Find the derivative of \( u \) with respect to \( x \)

For \( u = 5x + 14 \), \( \frac{du}{dx} = 5 \).

Step3: Multiply the two derivatives

Using the chain rule, \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^{4} \cdot 5 \). Substitute back \( u = 5x + 14 \): \( \frac{dy}{dx} = 5(5x + 14)^{4} \cdot 5 = 25(5x + 14)^{4} \).

Answer:

\( 25(5x + 14)^{4} \)