# TECHNIQUES OF DIFFERENTIATION

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TECHNIQUES OF

DIFFERENTIATION

BY

EDSEL M. LLAVE

TECHNIQUES OF DIFFERENTIATION

• INTRODUCTION

• Two fundamental problems:

– the tangent problem

– the area problem

• The portion of calculus arising from the tangent problem

such as given a function f and a point P(x0, y0) on its graph,

find the equation of the line tangent to the graph at P is

called differential calculus

• That arising from the area problem which is stated as given

the function f, find the area between the graph of f and an

interval [a, b] on the x-axis is called integral calculus.

TECHNIQUES OF DIFFERENTIATION

• In plane geometry, a line is called tangent to a circle if it

meets the circle at a precisely one point. However, this

definition is not satisfactory for other kinds of curves.

• To define the concept of a tangent line so that it applies to

curves other than circles, let us consider a point P on a

curve in the xy-plane.

• If Q is any point on the curve different from P, the line

through P and Q is called a secant line for the curve.

• Intuition suggests that if we move the point Q along the

curve toward P, the secant line will rotate toward a

“limiting” position.

• The line T occupying this limiting position is said to be the

tangent line at P.

TECHNIQUES OF DIFFERENTIATION

• If P(x0, y0) and Q(x1, y1) are distinct points on

such a curve, then the secant line connecting

P and Q has slope

TECHNIQUES OF DIFFERENTIATION

• If we let x1 approach x0, then Q will approach

P along the graph of f, and the secant line

through P and Q will approach the tangent

line at P.

• Thus, the slope msec of the secant line

approaches the slope mtan of the tangent line

as x1 approaches x0.

• Therefore,

TECHNIQUES OF DIFFERENTIATION

TECHNIQUES OF DIFFERENTIATION

• Definition 1

• If P(x0, y0) is a point on the graph of a function

f, then the tangent line to the graph of f at P is

defined to be the line through P with slope

• Provided this limit

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