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

1 Answer
Jim H
May 22, 2015

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#.

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