# Derivative using definition

• Sep 27th 2009, 11:48 PM
Latszer
Derivative using definition
If f(x)=1/x^2 find f'(x) using the definition of the derivative.
(note: If you instead use the power rule, f'(x)=nx^(n-1), you will get zero credit for this problem. It is useful check of your answer though.)

The answer is -2/x^3 and the best I can get is -1/x^3.
Could someone get me on the right track? If you can't help by tomorrow, don't bother because it is due tomorrow.

Thanks!
• Sep 27th 2009, 11:56 PM
Prove It
Quote:

Originally Posted by Latszer
If f(x)=1/x^2 find f'(x) using the definition of the derivative.
(note: If you instead use the power rule, f'(x)=nx^(n-1), you will get zero credit for this problem. It is useful check of your answer though.)

The answer is -2/x^3 and the best I can get is -1/x^3.
Could someone get me on the right track? If you can't help by tomorrow, don't bother because it is due tomorrow.

Thanks!

$\displaystyle f(x) = \frac{1}{x^2}$

$\displaystyle f(x + h) = \frac{1}{(x + h)^2}$

$\displaystyle f(x + h) - f(x) = \frac{1}{(x + h)^2} - \frac{1}{x^2}$

$\displaystyle = \frac{x^2}{x^2(x + h)^2} - \frac{(x + h)^2}{x^2(x + h)^2}$

$\displaystyle = \frac{x^2 - (x + h)^2}{x^2(x + h)^2}$

$\displaystyle = \frac{x^2 - x^2 - 2xh - h^2}{x^2(x + h)^2}$

$\displaystyle = \frac{h(-2x - h)}{x^2(x + h)^2}$.

$\displaystyle \frac{f(x + h) - f(x)}{h} = \frac{h(-2x - h)}{hx^2(x + h)^2}$

$\displaystyle = \frac{-2x - h}{x^2(x + h)^2}$.

$\displaystyle \lim_{h \to 0}\frac{f(x + h) - f(x)}{h} = \lim_{h \to 0}\frac{-2x - h}{x^2(x + h)^2}$

$\displaystyle = \frac{-2x}{x^2(x^2)}$

$\displaystyle = -\frac{2x}{x^4}$

$\displaystyle = -\frac{2}{x^3}$.

Thus $\displaystyle f'(x) = -\frac{2}{x^3}$.
• Sep 28th 2009, 12:06 AM
Latszer
Thanks man, I factored wrong the first time I did it, I understand now.