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 ${f}^{\prime }\left(0\right)=-\frac{2}{{3}^{2}}=-\frac{2}{9}$
  
  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.
  
  Didn't find what you were looking for?
explanation: The document contains a math question and a detailed answer explaining the solution. The question asks for the instantaneous rate of change of the function \( f(x) = \frac{1}{{x}^{2} + 2x + 3} \) at \( x = 0 \). The answer provides a step-by-step explanation using the chain rule to find the derivative and then evaluates it at \( x = 0 \).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 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.
  
  Didn't find what you were looking for?
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer explaining the solution. The question asks for the instantaneous rate of change of the function \( f(x) = \frac{1}{{x}^{2} + 2x + 3} \) at \( x = 0 \). The answer provides a step-by-step explanation using the chain rule to find the derivative and then evaluates it at \( x = 0 \).
  
  Label: question-and-answer