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How do you find the derivative of $f (x) = 9 - x^{2}$ $f(x)= 9-x^2$ using the limit definition at x=2?

1 Answer

May 16, 2017

$f'(2) = -4$

Explanation:

By definition $f'(x) = lim_"h->0" (f(x+h)-f(x))/h$

In our example $f(x) = 9-x^2$

Hence $f'(x) = lim_"h->0" (9-(x+h)^2-(9-x^2))/h$

$= lim_"h->0" (9-x^2-2xh-h^2-9+x^2)/h$

$= lim_"h->0"(-2xh-h^2)/h$

$= lim_"h->0"-2x-h = -2x$

$:. f'(2) = -2*2 = -4$

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