Question

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

Answer 1

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

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

Method 1

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

Good! Make it so.

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

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

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

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

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)`

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

In either case, to find the derivative, evaluate the limit as `h rarr 0`.