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

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

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Answer 1 · 2016-03-10T14:49:43

$- \frac{2}{9}$ $-2/9$

Explanation:

This is just the value of f'(0)

$f (x) = \frac{1}{x^{2} + 2 x + 3} = {(x^{2} + 2 x + 3)}^{- 1}$ $f(x) = 1/(x^2+2x+3) = (x^2+2x+3)^-1$

using the $chain rule$ $color(blue) " chain rule "$

$d/dx[f(g(x))] = f'(g(x)). g'(x)$

$rArr f'(x) = - (x^2+2x+3)^-2 d/dx(x^2+2x+3)$

$= -(2x+2)/(x^2+2x+3)^2$

and f'(0) = $- 2/(3^2) = -2/9$

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

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Explanation:

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