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

1 Answer
Nov 24, 2016

Please see the explanation.

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#