#### Transcript AP Calculus BC – 3.6 Chain Rule 1

AP Calculus BC – Chapter 3: Derivatives 3.1: Derivative of a Function Goals: Calculate slopes and derivatives using the definition of derivative.

Graph f from the graph of f’, graph f’ from the graph of f, and graph the derivative of a function given numerically.

Quote for today: “That which we must learn to do, we learn by doing.” Aristotle (382 BC – 322 BC), in Nicomachean Ethics

Definition of Derivative: The derivative of the function f with respect to the variable x is the function f’ whose value at x is

*f*

*h*

lim 0

*f*

*x*

*h h*

provided the limit exists.

(Alternate) Definition of Derivative at a Point: The derivative of the function f at the point x=a is the limit

*f*

*x*

lim

*a f x*

*a*

provided the limit exists.

Differentiable: The domain of f’, the set of points in the domain of f for which the limit exists, may be smaller than the domain of f. If f’(x) exists, we say that f has a derivative at x (or is differentiable at x). A function that is differentiable at every point of its domain is a differentiable function.

One-Sided Derivatives: 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

*h*

lim 0

*f*

*a*

*h h*

right-hand limit

*f*

*b*

*h*

*h*

lim 0

*h*

left-hand limit exist at the endpoints.

Assignment: HW 3.1: #1, 3, 6, 9, 12-15, 17, 20, 26.

Quiz after 3.3.