Question

In the end of lecture 2, the professor proofs the theorem (DIFF->CTS) and states that (f(x)-(fx0))/(x-x0) = f'(x0). Can anyone clarify this for me? What's the relationship between that first term and the derivative of f(x0) ? Thanks

Answer 1

This question pertains to the proof that differentiability implies continuity, which appears in Clip 5 of Session 5.

From the first lecture, we know that the derivative at x0 is the limit as delta x goes to zero of\[\frac{ f(x_0+\Delta x)-f(x_0) }{ \Delta x }\]That expression is known as the difference quotient, and we know that if a function is differentiable, the limit as delta x goes to zero must exist, because that limit is the derivative.

The left side of the equation in your message is just an alternative way of writing the difference quotient. You'll see this if you recall that delta x is shorthand for (x-x0).