# How do I find the derivative of 1/x using the difference quotient?

May 22, 2015

The crucial bit of algebra (and the one you're probably stuck on) is:

$\frac{\frac{1}{x + h} - \frac{1}{x}}{h}$

Method 1

"If I had a fraction over a fraction, I'd know what to do next."

Good! Make it so.

$\frac{\frac{1}{x + h} - \frac{1}{x}}{h} = \frac{\frac{x}{x \left(x + h\right)} - \frac{x + h}{x \left(x + h\right)}}{\frac{h}{1}}$

$= \frac{\frac{x - \left(x + h\right)}{x \left(x + h\right)}}{\frac{h}{1}}$

$= \frac{- h}{x \left(x + h\right)} \cdot \frac{1}{h}$

$= \frac{- 1}{x \left(x + h\right)}$

Method 2

"I know this trick:"
Multiply numerator and denominator by the common denominator of all the fractions in the numerator and denominator. (Sounds complicated, but look:)

(1/(x+h) - 1/x)/h = ((1/(x+h) - 1/x))/h (x(x+h))/(x(x+h)

= ((x(x+h))/(x+h) - (x(x+h))/x)/(h(x(x+h))

=(x-(x+h))/(hx(x+h)

$= \frac{- 1}{x \left(x + h\right)}$

In either case, to find the derivative, evaluate the limit as $h \rightarrow 0$.