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Using the limit definition, how do you differentiate $f (x) = 1 - x^{2}$ $f(x)=1-x^2$ ?

1 Answer

Nov 3, 2015

If $f (x) = 1 - x^{2}$ $f(x) = 1-x^2$ then $\frac{d f (x)}{d x} = - 2 x$ $(d f(x))/(dx) = -2x$

Explanation:

The limit definition of the derivative of a function $f (x)$ $f(x)$ is
$color(white)("XXX")(d f(x))/(dx) = lim_(hrarr0) (f(x+h)-f(x))/h$

If $f(x) = 1-x^2$
$color(white)("XXX")$ then (by the limit definition)
$(d f(x))/(dx)$
$color(white)("XXX")=lim_(hrarr0) ((1-(x+h)^2)-(1-x^2))/h$

$color(white)("XXX")= lim_(hrarr0) (1-x^2-2xh-h^2-1+x^2)/h$

$color(white)("XXX")=lim_(hrarr0)(-2xh-h^2)/h$

$color(white)("XXX")=lim_(hrarr0) -2x-h$

$color(white)("XXX")=-2x$

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