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