# How do you find f'(x) using the definition of a derivative for #f(x)= (x^2-1) / (2x-3)#?

Long divide then use limit definition of derivative to find:

#f'(x) = 1/2 - 5/(2(2x-3)^2)#

Long divide:

...to find...

#f(x) = (x^2-1)/(2x-3)=1/2x+3/4+5/(8x-12)#

Then using the limit definition of derivative:

#f'(x) = lim_(h->0) ((f(x+h) - f(x))/h)#

#=lim_(h->0) ((1/2(x+h)+3/4+5/(8(x+h)-12)) - (1/2x+3/4+5/(8x-12)))/h#

#=lim_(h->0) (1/2h+(5/(8(x+h)-12)-5/(8x-12)))/h#

#=1/2+lim_(h->0) ((5/(8(x+h)-12)-5/(8x-12))/h)#

#=1/2+lim_(h->0) ((5((8x-12)-(8(x+h)-12)))/(h(8(x+h)-12)(8x-12)))#

#=1/2+lim_(h->0) ((-40h)/(h(8(x+h)-12)(8x-12)))#

#=1/2+lim_(h->0) ((-40)/((8(x+h)-12)(8x-12)))#

#=1/2-40/((8x-12)^2)#

#=1/2-40/(16(2x-3)^2)#

#=1/2-5/(2(2x-3)^2)#

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To find ( f'(x) ) using the definition of a derivative for ( f(x) = \frac{{x^2 - 1}}{{2x - 3}} ), you apply the definition of the derivative:

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

First, substitute ( f(x) ) into the definition:

[ f'(x) = \lim_{{h \to 0}} \frac{{\frac{{(x + h)^2 - 1}}{{2(x + h) - 3}} - \frac{{x^2 - 1}}{{2x - 3}}}}{h} ]

Next, simplify the expression:

[ f'(x) = \lim_{{h \to 0}} \frac{{(x^2 + 2xh + h^2 - 1)(2x - 3) - (x^2 - 1)(2(x + h) - 3)}}{{h(2x - 3)(2(x + h) - 3)}} ]

[ f'(x) = \lim_{{h \to 0}} \frac{{2x^3 + 2x^2h + 2x^2h + 2xh^2 - 3x^2 - 4xh + 3 - 2x^3 - 2xh + 3x^2 + 2h - 3}}{{h(2x - 3)(2x + 2h - 3)}} ]

[ f'(x) = \lim_{{h \to 0}} \frac{{4xh^2 - 4xh + 2h}}{{h(2x - 3)(2x + 2h - 3)}} ]

[ f'(x) = \lim_{{h \to 0}} \frac{{2h(2x - 3)}}{{h(2x - 3)(2x + 2h - 3)}} ]

Cancel out ( h ) in the numerator and denominator:

[ f'(x) = \lim_{{h \to 0}} \frac{{2}}{{2x + 2h - 3}} ]

[ f'(x) = \frac{{2}}{{2x - 3}} ]

So, ( f'(x) = \frac{{2}}{{2x - 3}} ).

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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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