Question

attempt 1: 10 attempts remaining. compute the derivative of the function $y = 0.5(7x^2 + 2x - 5)^4$ using the chain rule. $\frac{dy}{dx} = $

Explanation:

Step1: Identify outer and inner functions

Let \( u = 7x^2 + 2x - 5 \) (inner function) and \( y = 0.5u^4 \) (outer function).

Step2: Differentiate outer function

Differentiate \( y \) with respect to \( u \): \( \frac{dy}{du} = 0.5 \times 4u^{3} = 2u^{3} \)

Step3: Differentiate inner function

Differentiate \( u \) with respect to \( x \): \( \frac{du}{dx} = 14x + 2 \)

Step4: Apply Chain Rule

Chain Rule: \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \). Substitute \( u = 7x^2 + 2x - 5 \), \( \frac{dy}{du} = 2(7x^2 + 2x - 5)^{3} \) and \( \frac{du}{dx} = 14x + 2 \):
\( \frac{dy}{dx} = 2(7x^2 + 2x - 5)^{3} \times (14x + 2) \)
Simplify the coefficient: \( 2\times(14x + 2)=4(7x + 1) \), so \( \frac{dy}{dx} = 4(7x + 1)(7x^2 + 2x - 5)^{3} \)

Answer:

\( 4(7x + 1)(7x^2 + 2x - 5)^3 \)