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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
image

or
image

exists.

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.
image
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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