Monday, June 20, 2011

6.1 – Differentiability of a Function

In Maths T, you already learnt how to prove whether a function is continuous. Now you need to know the relationship between continuity and differentiability.

A differentiable function has to be continuous, but it doesn’t mean that a continuous function is differentiable. Using logical propositions, it means that if f(x) differentiable, then it is continuous, but not conversely. Normally, the non-differentiability occurs in graphs with

1. a corner                                                                     2. a vertical tangent line
image                image

3. a discontinuity                                                      4. at end points
image           image

For piece-wise defined functions, it is easy to see whether a function is differentiable at the joints. If the joints have different gradients for the different sub-functions, then it is definitely not differentiable. However, there should be a formal definition for differentiability. For a number a in the domain of the function f, we say that f is differentiable at a , or that the derivatives of f exists at a if



You can go on to prove that both formulas are actually the same thing. Of course, differentiability does not restrict to only points. We could also say that a function is differentiable on  an interval (a, b) or differentiable everywhere, (-∞, +∞). I’ll give you one example:

Prove that f(x) = |x| is not differentiable at x=0.
So, f(x) = |x| is not differentiable at x = 0. [proven]

These 2 formulas can be used at different situations, so if one doesn’t work, use the other. Differentiability is not a common question in STPM, but you should still be able to make use of this important information.

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