Tangent lines n Recall tangent line is the

Tangent lines n Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a, f(a)) is the line through P with slope n 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)). n

Tangent lines n Ex. 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 is or simplifies to

Velocities n n Recall: instantaneous velocity is limit of average velocity Suppose 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 change n n Let quotient The difference 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)=t 3 -t+1. What is the rate of change of temperature with respective to time at t=2? Sol. The rate of change is n

Definition of derivative Definition The derivative of a function f at a number a, denoted by is n if the limit exists. Similarly, we can define left-hand derivative exists if and only if both and they are the same. and rightexist and

Example n Ex. Find the derivative given Sol. Since the derivative does not exist, does not exist.

Example Ex. Determine the existence of Sol. Since n does not exist. of f(x)=|x|.

Continuity and derivative n Theorem If exists, then f(x) is continuous at x 0. Proof. n Remark. The continuity does not imply the existence of derivative. For example,

Interpretation of derivative n n The 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 function n n Recall 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 derivative exists, we can regard as a function which assigns to x.

Remark n Some other limit forms

Example Find the derivative function of Sol. Let a be any number, by definition, Letting a vary, we get the derivative function

Other notations for derivative n If 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. n 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.