How do you use the definition of a derivative to find the derivative of #f(x) = -7x^2 + 4x#?

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Answer 1 · 2016-08-20T13:26:05

#f'(x)=-14x+4#

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

In this example: #f(x) = -7x^2+4x#

Therefore: #f'(x) = lim_"h->0" (-7(x+h)^2+4(x+h) -(-7x^2+4x))/h#

#= lim_"h->0" (-7(x^2+2xh+h^2)+4x+4h+7x^2-4x)/h#

#= lim_"h->0" (-7x^2-14xh-7h^2+4x+4h+7x^2-4x)/h#

#= lim_"h->0" (cancel(-7x^2)-14xh-7h^2cancel(+4x)+4hcancel(+7x^2)cancel(-4x))/h#

#=lim_"h->0" (-14xh-7h^2+4h)/h#

#=lim_"h->0" -14x-7h+4#

#= -14x+4#