How do you find the derivative of #f(x)=1-x^2# using the limit process?

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Answer 1 · 2017-02-03T16:06:56

#d/(dx) (1-x^2) = -2x#

By definition the derivative is:

#f'(x) = lim_(h->0) (f(x+h)-f(x))/h#

Evaluate the limit in the case: #f(x) = 1-x^2#:

#d/(dx) (1-x^2) = lim_(h->0) ((1-(x+h)^2-(1-x^2))/h)#

#:.d/(dx) (1-x^2) = lim_(h->0) (cancel(1)-cancel(x^2)-2hx-h^2cancel(-1)+cancel(x^2))/h#

#:.d/(dx) (1-x^2) = lim_(h->0) (-2hx-h^2)/h = lim_(h->0) (cancel(h)(-2x-h))/cancel(h)#

#:.d/(dx) (1-x^2) = lim_(h->0) (-2x-h) = -2x#