How do you find the derivative using limits of #f(x)=1/(x-1)#?

Answer 1

Please see the explanation.

Given:#f(x) = 1/(x - 1)#, then #f(x + h) = 1/(x + h - 1)#

Find f'(x)

#f'(x) = lim_(hto0){f(x + h) - f(x)}/h#
Substitute #1/(x + h - 1)# for #f(x + h)# and #1/(x - 1)# for #f(x)#:
#f'(x) = lim_(hto0){1/(x + h - 1) - 1/(x - 1)}/h#
Multiply by 1 in the form of #(x - 1)/(x - 1)#:
#f'(x) = lim_(hto0){1/(x + h - 1) - 1/(x - 1)}/h(x - 1)/(x - 1)#

Multiply numerators and denominators:

#f'(x) = lim_(hto0){(x - 1)/(x + h - 1) - (x - 1)/(x - 1)}/(h(x - 1))#

The second term in the numerator becomes 1:

#f'(x) = lim_(hto0){(x - 1)/(x + h - 1) - 1}/(h(x - 1))#
Multiply by 1 in the form of #(x + h - 1)/(x + h - 1)#:
#f'(x) = lim_(hto0){(x - 1)/(x + h - 1) - 1}/(h(x - 1))(x + h - 1)/(x + h - 1)#

Multiply numerators and denominators:

#f'(x) = lim_(hto0){((x - 1)(x + h - 1))/(x + h - 1) - (x + h - 1)}/(h(x - 1)(x + h - 1))#

I shall mark what cancels:

#f'(x) = lim_(hto0){((x - 1)cancel(x + h - 1))/cancel(x + h - 1) - (x + h - 1)}/(h(x - 1)(x + h - 1))#

Remove the canceled factors:

#f'(x) = lim_(hto0){(x - 1) - (x + h - 1)}/(h(x - 1)(x + h - 1))#

Distribute the -1 in the numerator:

#f'(x) = lim_(hto0){x - 1 - x - h + 1}/(h(x - 1)(x + h - 1))#

Combine like terms in the numerator:

#f'(x) = lim_(hto0){-h}/(h(x - 1)(x + h - 1))#
#-h/h# becomes -1:
#f'(x) = lim_(hto0){-1}/((x - 1)(x + h - 1))#
It is safe to let #hto0#:
#f'(x) = {-1}/((x - 1)(x - 1))#

Simplify:

#f'(x) = -1/(x - 1)^2#
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Answer 2

To find the derivative of ( f(x) = \frac{1}{x - 1} ) using limits, you apply the definition of the derivative:

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

Substitute ( f(x) = \frac{1}{x - 1} ) into the formula and simplify.

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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.

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.

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.

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.

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