Charity Davies

2022-03-14

What is the instantaneous rate of change of $f\left(x\right)=\frac{1}{{x}^{2}+2x+3}$ at x=0?

### Answer & Explanation

IdodaHekbed7mx

Explanation:
This is just the value of f'(0)
$f\left(x\right)=\frac{1}{{x}^{2}+2x+3}={\left({x}^{2}+2x+3\right)}^{-1}$
using the chain rule
$\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]={f}^{\prime }\left(g\left(x\right)\right)\cdot {g}^{\prime }\left(x\right)$
$⇒{f}^{\prime }\left(x\right)=-{\left({x}^{2}+2x+3\right)}^{-2}\frac{d}{dx}\left({x}^{2}+2x+3\right)$
$=-\frac{2x+2}{{\left({x}^{2}+2x+3\right)}^{2}}$
and ${f}^{\prime }\left(0\right)=-\frac{2}{{3}^{2}}=-\frac{2}{9}$

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Charity Davies

2022-03-14

What is the instantaneous rate of change of $f\left(x\right)=\frac{1}{{x}^{2}+2x+3}$ at x=0?

### Answer & Explanation

IdodaHekbed7mx

Explanation:
This is just the value of f'(0)
$f\left(x\right)=\frac{1}{{x}^{2}+2x+3}={\left({x}^{2}+2x+3\right)}^{-1}$
using the chain rule
$\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]={f}^{\prime }\left(g\left(x\right)\right)\cdot {g}^{\prime }\left(x\right)$
$⇒{f}^{\prime }\left(x\right)=-{\left({x}^{2}+2x+3\right)}^{-2}\frac{d}{dx}\left({x}^{2}+2x+3\right)$
$=-\frac{2x+2}{{\left({x}^{2}+2x+3\right)}^{2}}$
and ${f}^{\prime }\left(0\right)=-\frac{2}{{3}^{2}}=-\frac{2}{9}$

Do you have a similar question?

Recalculate according to your conditions!

Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

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The text contains a mathematical question about finding the instantaneous rate of change of a function at a specific point, and it provides a detailed explanation and answer to the question.

TYPE: SINGLE_QUESTION_WITH_ANSWER