Help with finding a derivative

KendraK

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Sep 20, 2015
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Hello, I have a homework assignment where I need to write my solution in such a way that someone else can follow along. Trouble is, I can't even figure out how to get the problem started (so, I probably can't get it finished either.)

The question/problem is:

Find the derivative or f(x)= 1/sqrt (x^2 + 3x -4) by the definition. In other words, use lim h-> 0 (f(x+h) -f(x))/h

Should I start by taking the denominator, moving it up with a -1/2 power and ??

The examples in class were not near as difficult. :/

Thanks for any help.
~Kendra
 
The question/problem is:

Find the derivative or f(x)= 1/sqrt (x^2 + 3x -4) by the definition. In other words, use lim h-> 0 (f(x+h) -f(x))/h

Should I start by taking the denominator, moving it up with a -1/2 power and ?? \(\displaystyle \ \ \ \ \)I wouldn't try that.

The examples in class were not near as difficult. :/

Rewrite the limit as:


\(\displaystyle \displaystyle\lim_{h \to \ 0}\ \dfrac{1}{h} \bigg( \dfrac{1}{\sqrt{(x + h)^2 \ + \ 3(x + h) \ - \ 4 \ }} \ - \ \dfrac{1}{\sqrt{x^2 + 3x -4}}\bigg) \ =\)


\(\displaystyle \displaystyle\lim_{h\to\ 0} \ {\frac{1}{h}}\cdot\dfrac{\sqrt{x^2 + 3x - 4} \ - \ \sqrt{(x + h)^2 \ + \ 3(x + h) \ - \ 4}}{(\sqrt{x^2 + 3x - 4} \ )\sqrt{(x + h)^2 \ + \ 3(x + h) \ - \ 4}}\)


Then multiply the numerator and the denominator by the conjugate of the numerator.




(I would never give this as a regular exercise, because it is so algebra-laden and time consuming.

I might give it as an extra credit assignment to do after class.)
 
...Then multiply the numerator and the denominator by the conjugate of the numerator.
Note: This step is the "trick". Be sure to include, in your "instructions", a note to the "reader" that s/he'll want to remember this, because it'll probably come up on the next test.

Then make sure you memorize it yourself! ;)
 
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