derivative

[madeenaa]

Find the derivative of the function y = 5/(X^2-9)
This is what I have so far: 5/(x+h)^2-9 = 5/(x^2+2xh+h^2-9)= ((5-(x^2-9))/(x^2+2xh+h^2-9))/h
I'm having a hard time solving this, please help...

 

[mmm4444bot]

At first, I thought you were trying to determine f(x + h), but then you subtracted stuff from it and divided it by stuff, so I can't follow all of your equal signs.

This part of your work is correct: f(x + h) = 5/[(x + h)^2 - 9]

This is just one of the terms in the Difference Quotient's numerator; you need to subtract f(x) from it.

[f(x + h) - f(x)]/h

The derivative of f(x) is the limit of this Difference Quotient as h goes to zero.

You should start by writing out the entire Difference Quotient, and then simplify that.

Since f(x) and f(x + h) are rational functions, we can avoid working with a compound ratio by first factoring out 1/h from the Difference Quotient.

(1/h) * [f(x + h)^2 - f(x)]

This is the same Difference Quotient. Okay ?

So, here's the Difference Quotient for function f:

\(\displaystyle \frac{1}{h} \cdot \left ( \frac{5}{(x + h)^2 - 9} - \frac{5}{x^2 - 9} \right )\)

Next, you could factor out a 5 from the numerators:

\(\displaystyle \frac{5}{h} \cdot \left ( \frac{1}{(x + h)^2 - 9} - \frac{1}{x^2 - 9} \right )\)

Can you continue ? Combine the two ratios inside the parentheses into a single ratio.

You don't need to multiply out the common denominator, but you do need to expand and simplify the numerator, in order to factor out an h to cancel with the bottom of 5/h.

After you cancel h's in the resulting factor 5h/h, take the limit as the remaining h's go to zero.

Let us know, if you have any specific questions about this.

 

[BigGlenntheHeavy]

\(\displaystyle madeenaa, \ after \ all \ is \ said \ and \ done, \ this \ reduces \ to:\)

\(\displaystyle f'(x) \ = \ \lim_{h\to0}\frac{-10x-5h}{[(x+h)^2-9][x^2-9]} \ = \ \lim_{h\to0}\frac{-10x}{[x^2-9][x^2-9]} \ = \ \frac{-10x}{[x^2-9]^2}\)

\(\displaystyle See \ the \ previous \ thread \ I \ did \ for \ you.\)

 

[madeenaa]

Thank you guys, I see what I was doing wrong.
BTW Glenn you have the best way of simplifying problems. Thank you for your help.