Summary O Level Additional Mathematics Chapter on Differentiation

CHAPTER 12: DIFFERENTIATION


Chapter objectives:

• understand the idea of a derived function
ð£ ðl £ ð ð£
• use the notations 𝑓 F (𝑥 ), 𝑓′′(𝑥), , H= x yI
ðT ðT l ðT ðT
•
• use the derivatives of the standard functions 𝑥 (for any rational 𝑛),
sin𝑥, cos 𝑥, tan 𝑥, 𝑒 T , ln 𝑥, together with constant multiples, sums
and composite functions of these
• differentiate products and quotients of functions
• apply differentiation to gradients, tangents and normals, stationary
points, connected rates of change, small increments and
approximations and practical maxima and minima problems
• use the first and second derivative tests to discriminate between
maxima and minima
• apply differentiation to kinematics problems that involve
displacement, velocity and acceleration of a particle moving in a
straight line with variable or constant acceleration, and the use of 𝑥– 𝑡
and 𝑣– 𝑡 graphs




Calculus

Calculus is the study of change in variables. Important techniques in calculus
include differentiation and integration.



What is differentiation?



226

,Differentiation is the process of determining the rate of change of a function
with respect to a variable at an instantaneous point. The derivative of a
function is its gradient function, so the value of the derivative at a point
represents the gradient of the graph at that point.

Given 𝑦 = 𝑓(𝑥) then the first derivative of 𝑦 with respect to 𝑥 equals the
first derivative of 𝑓(𝑥) which is written in the following notation:

𝑑𝑦
= 𝑓 F (𝑥)
𝑑𝑥
ð£
Where is the rate of change in 𝑦 with respect to 𝑥 (pronounced 𝑑𝑦 by 𝑑𝑥)
ðT
and 𝑓 F (𝑥) is the first derivative of 𝑓(𝑥) and is pronounced as such.



Basic rules of differentiation

1. If a function is the sum or difference of other functions, then the
derivative is also the sum or difference of the derivatives of those
functions i.e.:

𝑦 = 𝑓(𝑥 ) + 𝑔(𝑥 ) + ℎ(𝑥 ) + ⋯
𝑑𝑦
→ = 𝑓 F (𝑥) + 𝑔F (𝑥) + ℎF (𝑥) + ⋯
𝑑𝑥

2. A constant that multiplies or divides a function also multiplies or divides
its derivative i.e.:

𝑦 = 𝑎𝑓 (𝑥 )
𝑑𝑦
→ = 𝑎𝑓 F (𝑥)
𝑑𝑥




227

,3. A constant does not change so its rate of change and hence derivative is
zero i.e.:

𝑦=𝑎
𝑑𝑦
→ =0
𝑑𝑥




DERIVATIVES OF STANDARD FUNCTIONS



Derivative of polynomials

The derivative of the polynomial of the form 𝑥 • where 𝑛 is constant real
number is given by

𝒅(𝒙𝒏 )
= 𝒏𝒙𝒏^𝟏
𝒅𝒙



{From this it follows that the derivative of a constant equals 0 because the
ð(6T K )
constant 𝑎 = 𝑎𝑥 k which gives = 𝑎 × 0 × 𝑥 k^/ = 0}
ðT




The derivative of a polynomial in the form (𝑎𝑥 + 𝑏)• is given by:

𝒅(𝒂𝒙 + 𝒃)𝒏
= 𝒂𝒏(𝒂𝒙 + 𝒃)𝒏^𝟏
𝒅𝒙




228

, Example 12.1
ð£
Given 𝑦 = 2𝑥 ] + 𝑥 0 + 3 find .
ðT




SOLUTION

𝑦 = 2𝑥 ] + 𝑥 0 + 3

𝑦 𝑑(𝑥 ]) 𝑑(𝑥 0 ) 𝑑(3)
→ =2 + +
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥
𝑑𝑦
→ = 2 × 3 × 𝑥 ]^/ + 2 × 𝑥 0^/ + 0
𝑑𝑥
𝑑𝑦
∴ = 6𝑥 0 + 2𝑥
𝑑𝑥




Example 12.2
ð£
Find given that 𝑦 = (3𝑥 + 11)0k .
ðT




SOLUTION

𝑦 = (3𝑥 + 11)0k
𝑑𝑦
→ = 3 × 20 × (3𝑥 + 11)0k^/
𝑑𝑥
𝑑𝑦
∴ = 60(3𝑥 + 11)/¯
𝑑𝑥


229