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Tangent lines 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 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)).

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Tangent lines 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

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Velocities 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

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Rates of change Let 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)=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

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Definition of derivative Definition 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 and they are the same.

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

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

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

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Interpretation of derivative 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.

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Derivative as a function 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.

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Remark Some other limit forms

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

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Other notations for derivative 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. 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.

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