Question

section 2.6: chain rule (homework)
score: 70/170 answered: 7/17
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question 8
0/10 pts 4 99 details
if $f(x)=(4x + 6)^{-3}$, find $f(x)$.
find $f(2)$.
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Explanation:

Step1: Identify inner and outer functions

Let $u = 4x + 6$, so $y = u^{-3}$.

Step2: Differentiate outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=-3u^{-4}$ using the power - rule $\frac{d}{du}(u^n)=nu^{n - 1}$.

Step3: Differentiate inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=4$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}$ and $\frac{du}{dx}$ we get $f^{\prime}(x)=-3(4x + 6)^{-4}\cdot4=-12(4x + 6)^{-4}$.

Step5: Find $f^{\prime}(2)$

Substitute $x = 2$ into $f^{\prime}(x)$: $f^{\prime}(2)=-12(4\times2 + 6)^{-4}=-12(8 + 6)^{-4}=-12\times14^{-4}=-\frac{12}{14^{4}}=-\frac{12}{38416}=-\frac{3}{9604}$.

Answer:

$f^{\prime}(x)=-12(4x + 6)^{-4}$; $f^{\prime}(2)=-\frac{3}{9604}$