Differentiation Mks

SM SAINS ALAM SHAH, K.L. PROGRAM MKS ADDITIONAL MATHEMATICS

BY :

PN. DING HONG ENG

Topics

Paper 1

Paper 2 B

A 03

04

05

06

07 03

5.

Indices and Logarithms

2 2 3 3 2

6.

Coordinate Geometry

2

7.

Statistics

8.

Circular Measures

2 1 1 1

04 05

06

1 1

07 03

04

1 1

05

06

C 07 03

04

05 06 07

1 1

1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 2 3 2 1 2 1

1 1 1 1

1 2

2 3

1 2

1 2

1 2

1 2

1 2

9.

Differentiation

10.

Solution of Triangles

1 1 1 1 1

11.

Index Number

1 1 1 1 1

Topics

Paper 1

Paper 2 B

A 03

04

05

05

06 03

04

05

06 03

04

C

06

05 03

04

5.

Indices dan Logarithms

2

6.

Geometry Coordinates

2

7.

Statistics

8.

Circular Measures

1

9.

Differentiation

2

10.

Solution of Triangles

1

1

1

11.

Index Number

1

1

1

2

3

3

1

1

1

1 1

1

1 1

1

1

1

2

3

1

2

2

1 2

1

1

1 1 2

1 2

1

1 2

1

1 3

06

DIFFERENTIATIO N The first derivative

Differentiate axn Addition/Subtractionofalgebraicterms

The second derivative

Product Rule, Quotient Rule

Differentiate Composite Function

APPLICATION OF DIFFERENTIATION Gradient of a curve Gradient of tangent Gradient of normal Equation of tangent Equation of normal

maximum and minimum value/point

The rate of change

Small changes and approximati on

CONCEPT OF DIFFERENTIATION y=f(x)

y=f(x) y2

y2

Q(x2, y2)

Q(x2, y2)

Q1

y1 0

P(x1, y1)

y1

x1

Gradient of chord

x2

=

y2 − y1 x2 − x1

P(x1, y1)

0

x1

Q2 x2

When point Q approaches point P (i.e x2

y2 − y1 δy = x2 − x1 δx

Then When x2 Then

x1)

x1, δ x

0

 y2 − y1  δ y dy = lim = lim   x2 → x1 x − x  2 1  δ x→0 δ x dx

Differentiation Technicques Differentiate axn (a) If y = a, a is a constant --(b) If y = ax, a is a constant--(c) If y= axn, a is a constant ---

dy =0 dx dy =a dx

dy = nax n −1 dx

(d) Differentiate Addition, Subtraction of algebraic terms. If , then

f ( x) = p( x) ± q ( x) f ' ( x) = p' ( x) ± q' ( x)

Differentiate Product/ Quotient of two Polynomials • (a) If y = uv, then

dy dv du =u +v dx dx dx u • (b) If y = , then v du dv v −u dy = dx 2 dx dx v

Differentiate Composite Function If y = f(u) and u = g(x), then, the composite function dy dy du = × dx du dx

or

d (ax+b)n = an(ax+b)n-1 dx

The Second Derivative 2

d y d dy = ( ) = f ' ' ( x ) 2 dx dx dx

Application of Differentiation 1. The gradient of the curve y= f(x) at a point is the derivative of y with respect to x, i.e. dy or f’(x). dx

2. The gradient of tangent at point A is the value of 3. (Gradient of normal) x (gradient of tangen) = -1 y normal

tangent

x

dy dx

at point A.

Equation of Tangent and Equation of Normal Equation of tangent at point (x1, y1) with gradient m is y – y 1 = m ( x – x1) Equation of normal at point (x1, y1) is y – y1 =

1 − m(

x – x 1)

Maximum and Minimum Point/Value dy At the turning point (stationary point), 2 dx =0 d y

dx (a)For maximum point

2

<0

2

d y dx 2

(b)For minimum point >0 y 0 + + + 0 x O

The Rate of Change If y = f(x),

then

dy dy dx = × dt dx dt

is the rate of change of y with respect to time, t

SMALL CHANGES AND APPROXIMATION • If y = f(x) and ∂y is a small change in y corresponding with ∂x , a small change in x, then

∂ y dy ≈ ∂ x dx dy ∂ y ≈ .∂ x dx