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# Differentiation

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### Differentiation

1. 1. DIFFERENTIATION BY : PN. DING HONG ENG SM SAINS ALAM SHAH, K.L. PROGRAM MKS ADDITIONAL MATHEMATICS
2. 2. SPM PAST YEAR QUESTIONS
3. 3. 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
4. 4. 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
5. 5. 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
6. 6. 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
7. 7. Differentiation Technicques <ul><li>Differentiate ax n </li></ul><ul><li>If y = a, a is a constant --- </li></ul><ul><li>If y = ax, a is a constant--- </li></ul><ul><li>If y= ax n , a is a constant --- </li></ul><ul><li>(d) Differentiate Addition, Subtraction of algebraic terms. </li></ul><ul><li>If , then </li></ul>
8. 8. Differentiate Product/ Quotient of two Polynomials <ul><li>(a) If y = uv, then </li></ul><ul><li>(b) If , then </li></ul>
9. 9. Differentiate Composite Function <ul><li>If y = f(u) and u = g(x), </li></ul><ul><li>then, the composite function </li></ul><ul><li>or </li></ul><ul><li>(ax+b) n = an(ax+b) n-1 </li></ul>
10. 10. The Second Derivative
11. 11. 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
12. 12. Equation of Tangent and Equation of Normal <ul><li>Equation of tangent at point (x 1 , y 1 ) with gradient m is </li></ul><ul><li>y – y 1 = m ( x – x 1 ) </li></ul><ul><li>Equation of normal at point (x 1 , y 1 ) is </li></ul><ul><li>y – y 1 = ( x – x 1 ) </li></ul>
13. 13. Maximum and Minimum Point/Value <ul><li>At the turning point (stationary point), = 0 </li></ul><ul><li>For maximum point < 0 </li></ul><ul><li>For minimum point > 0 </li></ul>y x - - - - - - + + + 0 0 O
14. 14. The Rate of Change <ul><li>If y = f(x), then </li></ul><ul><li>is the rate of change of y with respect to time, t </li></ul>
15. 15. SMALL CHANGES AND APPROXIMATION <ul><li>If y = f( x ) and is a small change in y corresponding with , a small change in x , then </li></ul>
16. 16. SCORE A in Additional Mathematics
17. 17. THANKS