Question

attempt 1: 10 attempts remaining. compute the derivative of the function ( y = \frac{25}{(x^2 + 2x + 4)^8} ) using the chain rule. ( \frac{dy}{dx} = ) submit answer next item

Explanation:

Step1: Identify outer and inner functions

Let \( u = x^2 + 2x + 4 \), so \( y = \frac{25}{u^8}=25u^{-8} \).

Step2: Differentiate outer function

Differentiate \( y \) with respect to \( u \): \( \frac{dy}{du}=25\times(-8)u^{-9}=-200u^{-9} \).

Step3: Differentiate inner function

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

Step4: Apply Chain Rule

By Chain Rule, \( \frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx} \). Substitute \( u = x^2 + 2x + 4 \), \( \frac{dy}{du}=-200u^{-9} \) and \( \frac{du}{dx}=2x + 2 \):

$$ \frac{dy}{dx}=-200(x^2 + 2x + 4)^{-9}\times(2x + 2)=-\frac{200(2x + 2)}{(x^2 + 2x + 4)^9}=-\frac{400(x + 1)}{(x^2 + 2x + 4)^9} $$

Answer:

\( \boldsymbol{-\dfrac{400(x + 1)}{(x^2 + 2x + 4)^9}} \)