Question

attempt 1: 10 attempts remaining. compute the derivative of the function ( y = 0.5(3x^2 + 2x - 4)^8 ) using the chain rule. ( \frac{dy}{dx} = ) submit answer next item

Explanation:

Step1: Identify the outer and inner functions

Let \( u = 3x^2 + 2x - 4 \), so the outer function is \( y = 0.5u^8 \).

Step2: Differentiate the outer function with respect to \( u \)

Using the power rule, \( \frac{dy}{du} = 0.5 \times 8u^{7} = 4u^{7} \).

Step3: Differentiate the inner function with respect to \( x \)

\( \frac{du}{dx} = \frac{d}{dx}(3x^2 + 2x - 4) = 6x + 2 \).

Step4: Apply the Chain Rule (\( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \))

Substitute \( u = 3x^2 + 2x - 4 \) back into \( \frac{dy}{du} \), then multiply by \( \frac{du}{dx} \):
\( \frac{dy}{dx} = 4(3x^2 + 2x - 4)^7 \times (6x + 2) \)
Simplify the coefficient: \( 4\times(6x + 2)=4\times2\times(3x + 1)=8(3x + 1) \)
So \( \frac{dy}{dx} = 8(3x + 1)(3x^2 + 2x - 4)^7 \)

Answer:

\( 8(3x + 1)(3x^2 + 2x - 4)^7 \)