text: 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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