3.2 - Differentiability

Definition of differentiability

Basics of differentiability

A function $y=f(x)$ is differentiable on a closed interval $[a,b]$ if it has a derivative at every interior point of the interval and if the limits ${\displaystyle \underset{h\to 0^{+}}{lim}\frac{f(a+h)-f(a)}{h}}$ (the right-hand limit) and ${\displaystyle \underset{h\to 0^{-}}{lim}\frac{f(b+h)-f(b)}{h}}$ (the left-handed limit) are equal.

What does it really mean?

• You can think of it like being “locally linear” – if you zoom in on the curve, will it look like a line?
• Basically, all changes in slope should take place gradually (like a quadratic curve) and not instantly (like an absolute value curve).

Differentiability implies continuity

Types of non-differentiability

These are the ways a function can be non-differentiable at a point:

• Corners: one sided derivatives differ. Left slope $\ne$ right slope, but they are both numbers. Example: $f(x)=|x|$
• Cusps: slopes of secants go to different infinities (one goes to $+\mathrm{\infty }$ and the other to $-\mathrm{\infty }$). Example: $f(x)=x^{2/3}$
• Vertical tangent: slopes of secant lines approach the same infinities (both go to either $+$ or $-$ $\mathrm{\infty }$). Example: $f(x)=x^{1/3}$
• A discontinuity. Example: a piecewise function.

The IVT Also Applies to Derivatives

Taking a derivative on a calculator

Just check out this video.

• The main idea is that calculators are wrong when they try to calculate derivatives at a point of non-differentiability.

Checking for differentiability

1. Check for continuity by checking limits of the original function
2. Check that the right and left handed limits of the derivatives are equal