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

##### 1 Answer

May 22, 2015

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

**Method 1**

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

Good! Make it so.

#= ((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:)

#= ((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