Introduction to differentiation
- 1. Block 1 Introduction to Differentiation
- 2. What is Being Learned? • What differentiation is • Rule to differentiate • Know what the derivative is.
- 3. Consider how quickly a flower grows the water level in a barrel decreasing due to a leak how fast a rocket goes All to do with rates of change
- 4. Distance Time On a graph, gradient measures rate of change gradient of a curve? Use tangent at required point
- 5. Finding THE Rule Sketch y = x2 x -3 -2 -1 0 1 2 3 y 9 4 1 0 1 4 9 Plot point at x = 1 Draw tangent Find gradient of tangent Repeat for x = 3 and x = -2
- 6. Finding THE Rule y = x2 x = 1 → m = 2 x = 3 → m = 6 x = -2 → m = -4
- 7. Finding THE Rule x m 1 2 3 6 -2 -4 Rule for y = x2 m = 2x
- 8. Finding THE Rule y = x3 x = 1 → m = 3 x = 2 → m = 12 x = 3 → m = 27
- 9. Finding THE Rule x m 1 3 2 12 3 27 Rule for y = x3 m = x2 X 3 = 3x2 X 3 X 3X 2 X 3X 3 X 1
- 10. Summarising y = x2 m = 2x y = x3 m = 3x2 y = x4 m = x 43 The gradient to the curve rule is known as The Derivative
- 11. Applying the Rule y = x8 derivative = 8x7 y = x11 derivative = 11x10 y = 5x3 derivative = 3 X 5x2 = 15x2
- 12. Applying the Rule y = 4x2 derivative = 8x1 (usually written 8x) y = 6x3 derivative = 18x2 Nasty y = 5x derivative = 5x0 = 5 same as 5x1
- 13. Even Nastier? y = 10 Derivative = 0 Easiest just to remember these last two as separate rules same as 10x0
- 14. About the Derivative • Has its own rule Gives you • Gradient of a curve (and the tangent to curve) • Rates of change More to come!
- 15. Derivative Symbols Depend on way equation is given Ex f(x) = 2x5 f/ (x) = 10x4
- 16. Derivative Symbols Depend on way equation is given Ex y = 5x3 dy /dx = 15x2
- 17. Differentiation THE rule if y = axn then dy /dx= nax n-1 Known as the derivative
- 18. Ex1. y = 6x3 dy /dx = 18x2 Ex2. f(x) = 8x4 f/ (x)= 32x3 Another way to write the derivative
- 19. Ex3. y = 6x dy /dx = 6 Ex4. y = 12 dy /dx = 0 Ex.s 3 and 4 apply to all others of the same type.
- 20. Derivative Calculations If f(x) = 3x2 – 8x + 12 , calculate f/ (3) f/ (x) = 6x – 8 f/ (3) = 6(3) – 8 = 10
- 21. Uses of the Derivative (so far) • Gives gradient of tangent to any curve • Gives rates of change
- 22. Derivative Calculations If f(x) = 2x3 + 7x – 4, calculate f/ (4) f/ (x) = 6x2 + 7 f/ (4) = 6(42 ) + 7 = 103 (very steep!!!!!!!!!!!!)
- 23. Key Question If f(x) = (x + 5)2 , calculate f/ (4) f(x) = (x + 5)(x + 5) = x2 + 10x + 25 f/ (x) = 2x + 10 f/ (4) = 2(4) + 10 = 18