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

Answer 1

Luke Alvarez

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#

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Answer 2

Paisley Johnson

To find the derivative of ( f(x) = -5x ), you can use the definition of the derivative, which involves taking the limit of the difference quotient as the interval approaches zero. The derivative of ( f(x) ), denoted as ( f'(x) ) or ( \frac{df}{dx} ), is equal to the limit as ( h ) approaches zero of ( \frac{f(x+h) - f(x)}{h} ). By substituting the function ( f(x) = -5x ) into this formula, you can find the derivative.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.