How do you find the derivative using limits of $f (x) = - 5 x$ $f(x)=-5x$ ?

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Answer 1 · 2016-11-24T23:18:00

Please see the explanation.

$f'(x) = lim_(hto0)(f(x +h) - f(x))/h$

given: $f(x) = -5x$

Then: $f(x + h) = -5(x + h) = -5x - 5h$

Substitute $-5x - 5h$ for $f(x + h)$

$f'(x) = lim_(hto0)(-5x - 5h - f(x))/h$

Substitute $-5x$ for $f(x)$

$f'(x) = lim_(hto0)(-5x - 5h - -5x)/h$

The x terms cancel

$f'(x) = lim_(hto0)(cancel(-5x) - 5h - cancel(-5x))/h$

The $h/h$ cancels:

$f'(x) = lim_(hto0)(-5cancelh)/cancelh$

$f'(x) = lim_(hto0) -5$

Let the limit go to zero:

$f'(x) = -5$

How do you find the derivative using limits of f(x)=-5xf(x)=−5xf(x)=-5x?