Question

Find the derivative using limits of (x^(2)-1)/(2x-3) Using the formula f(x)-f(a)/x-a Thanks!!!

Answer 1

gotta do it by hand huh?

Answer 2

this is going to be a raft of algebra
we can to it
ready?

Answer 3

lets forget about the \(x-a\) in the denominator, and take care of that last

Answer 4

\[\frac{x^2-1}{2x-3}-\frac{a^2-1}{2a-3}\] is a start
now we actually have to subtract
\[\frac{(x^2-1)(2a-3)-(a^2-1)(2x-3)}{(2x-3)(2a-3)}\]

Answer 5

bunch of algebra at this step, which is why this is really annoying, but when you are done you should end up with
\[-\frac{(a-x) (a (2 x-3)-3 x+2))}{(2 a-3) (2 x-3)}\]

Answer 6

finally divide by \(x-a\) i.e. cancel

Answer 7

I just got stuck on where to go from (2a-3)(x^(2)-1)-(a^(2)-1)(2x-3) ??

Answer 8

the numerator is
\[ (2a-3)(x^2-1)-(a^2-1)(2x-3) \]
multiply it out to get
\[ 2ax^2 -3x^2 -2a + 3 - (2a^2x -3a^2 -2x +3) \\
2ax^2-3x^2 -2a + 3 -2a^2x + 3a^2 +2x -3
\]
now re-arrange and group terms (we are hoping to make form (x-a) factors)
\[ (2ax^2-2a^2x) -3x^2+ 3a^2+ (2x-2a) + (3 -3) \\
(2ax^2-2a^2x) - 3(x^2- a^2) + (2x-2a)
\]
now factor each term
\[ 2ax(x-a) - 3(x+a)(x-a) + 2(x-a) \]
and finally
\[ (2ax -3(x+a) +2) (x-a) \]

can you finish?

Answer 9

Yes!! Thank you. :)