Tangent linesRecall: tangent line is the limit of secant lineThe tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope

provided that the limit exists.Remark. If the limit does not exist, then the curve does not have a tangent line at P(a,f(a)).

Tangent linesEx. Find an equation of the tangent line to the hyperbola y=3/x at the point (3,1). Sol. Since the limit

an equation of the tangent line isor simplifies to

Velocities Recall: instantaneous velocity is limit of average velocitySuppose the displacement of a motion is given by the function f(t), then the instantaneous velocity of the motion at time t=a is

Ex. The displacement of free fall motion is given by find the velocity at t=5.Sol. The velocity is

Rates of changeLet The difference quotient

is called the average rate of change of y with respect to x. Instantaneous rate of change =

Ex. The dependence of temperature T with time t is given by the function T(t)=t3-t+1. What is the rate of change of temperature with respective to time at t=2?Sol. The rate of change is

Definition of derivativeDefinition The derivative of a function f at a number a,denoted by is

if the limit exists.

Similarly, we can define left-hand derivative and right-hand derivative exists if and only if both and exist andthey are the same.

ExampleEx. Find the derivative given

Sol. Since does not exist,

the derivative does not exist.

ExampleEx. Determine the existence of of f(x)=|x|. Sol. Since

does not exist.

Continuity and derivativeTheorem If exists, then f(x) is continuous at x0.

Proof.

Remark. The continuity does not imply the existence of derivative.

For example,

Interpretation of derivativeThe slope of the tangent line to y=f(x) at P(a,f(a)), is the derivative of f(x) at a,

The derivative is the rate of change of y=f(x) with respect to x at x=a. It measures how fast y is changing with x at a.

Derivative as a functionRecall that the derivative of a function f at a number a is given by the limit:

Let the above number a vary in the domain. Replacing a by variable x, the above definition becomes

If for any number x in the domain of f, the derivativeexists, we can regard as a function which assigns to x.

RemarkSome other limit forms

ExampleFind the derivative function of

Sol. Let a be any number, by definition,

Letting a vary, we get the derivative function

Other notations for derivativeIf we use y=f(x) for the function f, then the following notations can be used for the derivative:

D and d/dx are called differentiation operators.

A function f is called differentiable at a if exists. f is differentiable on [a,b] means f is differentiable in (a,b) and both and exist.