Differentiability Arches National Park- Park Avenue

North Window Arch

Balanced Rock

Delicate Arch

In General! Specific!

To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity

Two theorems: If f has a derivative at x = a, then f is continuous at x = a. Since a function must be continuous to have a derivative, if it has a derivative then it is continuous. **The converse of this theorem is false!

Intermediate Value Theorem for Derivatives Between a and b, must take on every value between and. If a and b are any two points in an interval on which f is differentiable, then takes on every value between and. 

If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: The derivative of a constant is zero. Rules for Differentiation Introduction

If we find derivatives with the difference quotient: (Pascal’s Triangle) We observe a pattern: …

examples: power rule We observe a pattern:…

examples: constant multiple rule: When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

(Each term is treated separately) constant multiple rule: sum and difference rules: This makes sense, because:

Example: Find the horizontal tangents of: Horizontal tangents occur when slope = zero. Plugging the x values into the original equation, we get: (The function is even, so we only get two horizontal tangents.)

First derivative (slope) is zero at: