Exploring Differentiability in Functions: A Comprehensive Guide

1. Section 2.2 Differentiability

2. Ways a function might not be differentiable. • 1. a corner • Often occurs with absolute value functions.

3. Ways a function might not be differentiable. • 2. A cusp. • Often seen when a function has a rational exponent… x2/3.

4. Ways a function might not be differentiable. • 3. A vertical tangent. • Example: Cube root function.

5. Ways a function might not be differentiable. • 4. A discontinuity.

6. Differentiable vs. Continuous • Differentiability implies local linearity. • A function starts to look like its tangent line when you zoom in very close. • If a function is differentiable, then it is continuous. The converse, however, is not necessarily true.

7. Derivatives on the Calculator • Math – 8 (nderiv) • Tell the calculator the variable, the function, and the value at which you are evaluating the derivative. You can enter x=x instead of a value if you want to graph the derivative.