Differentiable vs. Continuous The process of finding the derivative of a function is called Differentiation. A function is called Differentiable at x if the derivative exists at x. A function is called Differentiable on interval (a,b) if it is differentiable at all points in the interval. THEOREM: If a function, f, is differentiable at x=c, then f is continuous at x=c. BUT…… If f is continuous at x=c, it may or may not be differentiable at x=c. (drastic change in slope, pointy, cusp, vertical tangent). Check out page 106, #81-86

Differentiation Rules (part 1) The Constant Rule: OR If Then The Power Rule: If n is a rational number, then the function is differentiable and: Examples: 1.) 2.) 3.)

Differentiation Rules (part 1) The Constant Multiple Rule: If f(x) is a differentiable function and c is a real number, then cf(x), is also a differentiable function and: Examples:

Differentiation Rules (part 1) The Sum and Difference Rules: The sum (or difference) of two differentiable functions f and g is itself differentiable and: Examples: 1.) 2.)

Differentiation Rules (part 1) OH YES, Trig Rules: AND Examples: 1.) 2.)

Comparison of Rates of Change Recall the Slope of the Secant Line: This gave an Average slope between two points Also called an Average Rate of Change between two points Recall the Slope of the Tangent Line: This gives an Instantaneous Rate of Change at one point Also gives the slope of the curve at the point of tangency Used to determine the behavior of f(x) along its domain

How do we use this???? (a) Find the derivative of (b) Use it to find the points where the tangent line is horizontal (c) Compare the average rate of change for f(x) on the interval [0, 3] to the instantaneous rate of change at