Cambridge Additional Mathematics

344 Introduction to differential calculus (Chapter 13)

EXERCISE 13D

1 Find, from first principles, the gradient function of:

a f(x)=x b f(x)=5 c f(x)=2x+5

2 Find

dy

dx

from first principles given:

a y=4¡x b y=x^2 ¡ 3 x c y=2x^2 +x¡ 1

3 Use the first principles formula f^0 (a) = lim

h! 0

f(a+h)¡f(a)

h

to find the gradient of the tangent to:

a f(x)=3x+5at x=¡ 2 b f(x)=5¡ 2 x^2 at x=3

c f(x)=x^2 +3x¡ 4 at x=3 d f(x)=5¡ 2 x¡ 3 x^2 at x=¡ 2

Differentiationis the process of finding a derivative or gradient function.

Given a function f(x), we obtain f^0 (x) bydifferentiating with respect tothe variablex.

There are a number of rules associated with differentiation. These rules can be used to differentiate more

complicated functions without having to use first principles.

Discovery 4 Simple rules of differentiation

In this Discovery we attempt to differentiate functions of the form xn, cxn wherecis a constant, and

functions which are a sum or difference of polynomial terms of the form cxn.

What to do:

1 Differentiate from first principles: a x^2 b x^3 c x^4

2 Consider the binomial expansion:

(x+h)n=

¡n

0

¢

xn+

¡n

1

¢

xn¡^1 h+

¡n

2

¢

xn¡^2 h^2 +::::+

¡n

n

¢

hn

=xn+nxn¡^1 h+

¡n

2

¢

xn¡^2 h^2 +::::+hn

Use the first principles formula f^0 (x) = lim

h! 0

f(x+h)¡f(x)

h

to find the derivative of f(x)=xn for x 2 Z+.

3aFind, from first principles, the derivatives of: i 4 x^2 ii 2 x^3

b By comparison with 1 , copy and complete: “If f(x)=cxn, then f^0 (x)=::::::”

4aUse first principles to findf^0 (x)for:

i f(x)=x^2 +3x ii f(x)=x^3 ¡ 2 x^2

b Copy and complete: “If f(x)=u(x)+v(x) then f^0 (x)=::::::”

E Simple rules of differentiation

Remember the

binomial expansions.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\344CamAdd_13.cdr Tuesday, 7 January 2014 9:52:45 AM BRIAN