Cambridge Additional Mathematics

342 Introduction to differential calculus (Chapter 13)

5aObserve the function f(x)=ex using the CD software.

b What is the gradient function f^0 (x)?

Consider a general function y=f(x) where A is the point (x,f(x)) and B is the point

(x+h,f(x+h)).

The chord AB has gradient=

f(x+h)¡f(x)

x+h¡x

=

f(x+h)¡f(x)

h

If we let B approach A, then the gradient of AB approaches

the gradient of the tangent at A.

So, the gradient of the tangent at the variable point (x,f(x)) is lim

h! 0

f(x+h)¡f(x)

h

This formula gives the gradient of the tangent to the curve y=f(x) at the point (x,f(x)) for any value

ofxfor which this limit exists. Since there is at most one value of the gradient for each value ofx, the

formula is actually a function.

Thederivative functionor simplyderivativeof y=f(x) is defined as

f^0 (x) = lim

h! 0

f(x+h)¡f(x)

h

When we evaluate this limit to find a derivative function, we say we aredifferentiating from first principles.

Example 3 Self Tutor

Use the definition of f^0 (x) to find the gradient function of f(x)=x^2.

f^0 (x) = lim

h! 0

f(x+h)¡f(x)

h

= lim

h! 0

(x+h)^2 ¡x^2

h

= lim

h! 0

x^2 +2hx+h^2 ¡x^2

h

= lim

h! 0

h(2x+h)

h

= lim

h! 0

(2x+h) fas h 6 =0g

=2x

D Differentiation from first principles

x

y

A

f(x + h) B

x x+h

f(x)

y = (x)f

h

O

1

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\342CamAdd_13.cdr Tuesday, 7 January 2014 2:36:07 PM BRIAN