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# Differentiation

## on Jun 19, 2009

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## DifferentiationPresentation Transcript

• DIFFERENTIATION BY : PN. DING HONG ENG SM SAINS ALAM SHAH, K.L. PROGRAM MKS ADDITIONAL MATHEMATICS
• SPM PAST YEAR QUESTIONS
• FORM 4 2 1 2 2 1 1 07 1 07 1 1 07 2 1 1 1 2 07 1 1 06 1 1 06 1 06 3 1 1 1 3 06 1 1 05 1 1 05 1 1 05 3 05 1 1 Index Number 11. 1 2 2 Differentiation 9. 1 1 1 1 Circular Measures 8. 1 04 04 1 1 04 1 03 C 1 03 B 03 04 03 A 2 1 Solution of Triangles 10. Statistics Coordinate Geometry Indices and Logarithms 7. 6. 5. 1 1 2 2 Paper 2 Paper 1 Topics
• FORM 4 2 1 2 06 06 06 3 1 1 1 3 06 1 1 05 1 1 05 1 05 3 05 1 1 Index Number 11. 1 2 2 Differentiation 9. 1 1 1 1 Circular Measures 8. 1 04 04 1 1 04 1 03 C 1 03 B 03 04 03 A 2 1 Solution of Triangles 10. Statistics Geometry Coordinates Indices dan Logarithms 7. 6. 5. 1 1 2 2 Paper 2 Paper 1 Topics
• DIFFERENTIATION The first derivative 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 approximation Differentiate ax n Addition /Subtraction of algebraic terms
• y=f(x) Q(x 2 , y 2 ) P(x 1 , y 1 ) 0 x 1 x 2 y 2 y 1 Gradient of chord = When point Q approaches point P (i.e x 2 x 1 ) Then When x 2 x 1 ,  x 0 Then y=f(x) Q(x 2 , y 2 ) P(x 1 , y 1 ) 0 x 1 x 2 y 2 y 1 Q 1 Q 2 CONCEPT OF DIFFERENTIATION
• Differentiation Technicques
• Differentiate ax n
• If y = a, a is a constant ---
• If y = ax, a is a constant---
• If y= ax n , a is a constant ---
• (d) Differentiate Addition, Subtraction of algebraic terms.
• If , then
• Differentiate Product/ Quotient of two Polynomials
• (a) If y = uv, then
• (b) If , then
• Differentiate Composite Function
• If y = f(u) and u = g(x),
• then, the composite function
• or
• (ax+b) n = an(ax+b) n-1
• The Second Derivative
• The gradient of the curve y= f(x) at a point is the derivative of y with respect to x, i.e. or f’(x). Application of Differentiation 1. The gradient of tangent at point A is the value of at point A. 2. (Gradient of normal) x ( gradient of tangen) = -1 3. x y tangent normal
• Equation of Tangent and Equation of Normal
• Equation of tangent at point (x 1 , y 1 ) with gradient m is
• y – y 1 = m ( x – x 1 )
• Equation of normal at point (x 1 , y 1 ) is
• y – y 1 = ( x – x 1 )
• Maximum and Minimum Point/Value
• At the turning point (stationary point), = 0
• For maximum point < 0
• For minimum point > 0
y x - - - - - - + + + 0 0 O
• The Rate of Change
• If y = f(x), then
• is the rate of change of y with respect to time, t
• SMALL CHANGES AND APPROXIMATION
• If y = f( x ) and is a small change in y corresponding with , a small change in x , then
• SCORE A in Additional Mathematics
• THANKS