Home > Calculus > Limit Definition of Derivative

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

Answer 1

Cameron Alexander

#=-4#.

The Derivative of a function #f : f(x)=-4x+2,# at #x# is denoted

by #f'(x),# and, is defined by,

#f'(x)=lim_(trarrx)(f(t)-f(x))/(t-x)#.

#:. f'(x)=lim_(trarrx) {(-4t+2)-(-4x+2)}/(t-x)#

#=lim_(trarrx) -4cancel((t-x))/cancel((t-x)#

#=lim_(trarrx) -4#

#=-4#.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Emily Ammerman

To find the derivative of ( f(x) = -4x + 2 ) using the limit definition of the derivative, we apply the following formula:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Substitute ( f(x) = -4x + 2 ) into the formula:

[ f'(x) = \lim_{h \to 0} \frac{(-4(x + h) + 2) - (-4x + 2)}{h} ]

Expand and simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{-4x - 4h + 2 + 4x - 2}{h} ] [ f'(x) = \lim_{h \to 0} \frac{-4h}{h} ] [ f'(x) = \lim_{h \to 0} -4 ]

As ( h ) approaches 0, the derivative ( f'(x) ) equals -4. Therefore, the derivative of ( f(x) = -4x + 2 ) is ( f'(x) = -4 ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.