How to differentiate y= x√(x-1) using the limit definition of derivative

Question

How to differentiate y= x√(x-1) using the limit definition of derivative?

Answer 1

Here is the limit definition of a derivative:

f '(x) = lim(h→0) of (f(x+h) - f(x))/h

There is a LOT of algebra in this. I encourage you to work through it ALL.

In this case, f(x + h) = (x + h)√(x + h - 1) so f(x + h) - f(x) = (x + h)√(x + h - 1) - x√(x - 1). To resolve this limit algebraically, we can multiply numerator and denominator by the conjugate of the numerator: