Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation

P1^

5

O

c

y

y = c

x

Figure 5.9

The line y = c has

gradient zero and so

d

d

y

x = 0.

EXAMPLE 5.2 For each of these functions of x, find the gradient function.

(i) y = x^5 (ii) z = 7 x^6 (iii) p = 11 (iv) f(x) = 3

x

SOLUTION

(i) d

d

y

x

= 5 x^4

(ii) ddzx=× 67 xx^55 = 42

(iii)

d

d

p

x=^0

(iv) f(x) = 3 x–1

⇒ f′(x) = (–1) × 3 x–2

= −

3

x^2

Sums and differences of functions

Many of the functions you will meet are sums or differences of simpler ones. For

example, the function (3x^2 + 4 x^3 ) is the sum of the functions 3x^2 and 4x^3.

To differentiate a function such as this you differentiate each part separately and

then add the results together.

EXAMPLE 5.3 Differentiate y = 3 x^2 + 4 x^3.

SOLUTION

d

d

y

x=+^61 xx^2

2

Note

This may be written in general form as:

y = f(x) + g(x) ⇒ ddyx = f′(x) + g′(x).

You many find it

easier to write x^1 as x–1.