Using the limit definition, how do you differentiate $f(x)=3-x^2$ ?

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Answer 1 · 2015-12-20T15:40:32

The limit definition of the derivative:

$lim_(hrarr0)(f(x+h)-f(x))/h$

We know that $f(x)=3-x^2$ and
$f(x+h)=3-(x+h)^2$

$=lim_(hrarr0)(3-(x+h)^2-(3-x^2))/h$

$=lim_(hrarr0)(3-(x^2+2hx+h^2)-3+x^2)/h$

$=lim_(hrarr0)(3-x^2-2hx-h^2-3+x^2)/h$

$=lim_(hrarr0)(-2hx-h^2)/h$

$=lim_(hrarr0)-2x-h$

Now we can plug in $0$ for $h$ .

$=-2x$

Thus, $f'(x)=-2x$ .

Using the limit definition, how do you differentiate f(x)=3-x^2f(x)=3-x^2?