Differentiation Concepts in Mathematics

Explore the concept of finding gradients of curves, understanding the patterns, and proving gradients mathematically. Learn notations for expressing gradients and tackle a practice problem to enhance your understanding.

• Mathematics
• Gradients
• Curves
• Notations
• Understanding

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1. C1 Chapter 7 Differentiation Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 13th October 2013

2. Gradient of a curve The gradient of the curve at a given point can be found by: 1. Drawing the tangent at that point. 2. Finding the gradient of that tangent.

3. How could we find the gradient? We want to find the gradient at the point A. x represents a small change in x and y represents a small change in y.

4. How could we find the gradient? Suppose we re finding the gradient of y = x2 at the point A(3, 9). Gradient = 6 ?

5. How could we find the gradient? For the curve y = x2, we find the gradient for these various points. Can you spot the pattern? Point (1, 1) (4, 16) (2.5, 6.25) (10, 100) Gradient 2 8 5 20 For y = x2, gradient = 2x? Let s prove it...

6. Proof that gradient of y = x2 is 2x Suppose we add some tiny value, h, to x. Then: The lim bit means what this expression approaches as h tends towards 0 (x + h, (x+h)2) ? ? (x, x2) y x ? The h disappears as h tends towards 0.

7. Further considerations (This slide is intended for Further Mathematicians only) You may be wondering why we couldn t just set h to be 0 immediately. Why did we have to expand out the brackets and simplify first? If h was 0 at this stage, we d have 0/0. This is known as an indeterminate form (i.e. it has no value!) We don t like indeterminate forms, and want to find some way to remove them. In this particular case, just expanding the numerator resolves the problem. There are 7 indeterminate forms in total: 0/0, 00, inf/inf, inf inf, 1inf, inf^0, and 0 x inf

8. Notation These are ways in which we can express the gradient. dy dx = 2x y = x2 Leibniz's notation ? f(x) = x2 f (x) = 2x Lagrange s notation ?

9. Your Turn: What is the gradient of y = x3? (x + h, (x+h)3) ? (x, x3) ? y x ?

10. Differentiating xn Can you spot the pattern? x2 x3 x4 x5 x6 y 3x2 4x3 5x4 6x5 dy/dx 2x ? If y = xn, then dy/dx = nxn-1 ? If y = axn, then dy/dx = anxn-1 In general, scaling y also scales the gradient y = x7 dy/dx = 7x6 y = x10 dy/dx = 10x9 ? ? y = 2x3 dy/dx = 6x2 f(x) = 2x3 f (x) = 6x2 ? ? f(x) = x2 + 5x4 f (x) = 2x + 20x3y = 3x1/2 dy/dx = 3/2 x-1/2 ? ?

11. Differentiating cx and c y What is the gradient of the line y = 3x? How could you show it using differentiation? y = 3x = 3x1 Then dy/dx = 3x0 = 3 ? x What is the gradient of the line y = 4? How could you show it using differentiation? y = 4 y = 4 = 4x0 Then dy/dx = 0x-1 = 0 ? x

12. Differentiating cx and c ?? ??= 3 ? = 3? ? ? ? = 0 ? ? ? = 7 ?? ??= 1 ? ? = ? ?? ??= 0 ? ? = 74

13. Exercises Be sure to use the correct notation for the gradient. Function Gradient Point(s) of interest Gradient at this point ? ? ? ? ? ? ? ? ? ? ? ? = 2?5 ? ? = 7?3 ? = 2? + ?2 ? ? = ?3+ ? 1+ 3 ? ? = ?4 ? = ?3+ 4 ? =1 3?3+ ?2 dy/dx = 10x4 (2, 64) 160 f (x) = 21x2 (3, 189) 189 dy/dx = 2 + 2x (4, 24) 10 f (x) = 3x2 x-2 (2, 11.5) 11.75 f (x) = 4x3 (2, 16) 32 ? ? dy/dx = 3x2 (3, 31), (-3, -23) 3,0 , 1,4 27 ?? ??= ?2+ 2? ? 3 3

14. Test your knowledge so far... Edexcel C1 May 2012 ?? ??= 15?2 8? 1 3+ 2 ? ?2? ??2= 30? 8 3? 2 ? 3

15. Turning more complex expressions into polynomials We know how to differentiate things in the form ???. Where possible, put expressions in this form. 1 2 ? ? ? 1 ?2= ? 2 ? 1 ?= ? 1 ? 2 1 + ? = ? 1 1 2 2+ ? ? ? ?2 ? ?2 ?3 3 2 = ? 5 2 ? 1 ? ? ?= ? ? ?2? 1 = ?3 ?2

16. Exercise 7E (Page 115) 1 Use standard results to differentiate. ?? ??= ? 1 ?? ??= ? 4 ?? ??= 6? 4+1 ?? ??=1 ?? ??= 3 + 6? 2 ?? ??= 3?2 2? + 2 ?? ??= 24? 8 + 2? 2 ? ? 2 a) y = 2 ? c) y = 1 3?3 2 ?3+ ? 3? + 2? 1 2? 2 ? ? ? ? ? e) y = 2 3 1 1 3? 2 f) y = h) y =3?2 6 j) y = ? ?2 ? + 2 y = 3? 2 2? ? 4? +1 l) ? Find the gradient of the curve with equation ? = ?(?) at the point A where: 2 1 1 4,2 ? ? c) ? ? = ? and A is at 4 ?2 and A is at 2,5 4 4 d) ? ? = 3?

17. Finding equations of tangents Find the equation of the tangent to the curve ? = ?2 when ? = 3. Function of gradient: ?? ??= ?? ? ? Gradient when ? = 3: ? = ? ? ?-value when ? = 3: ? = ? ? = 3 So equation of tangent: ? ? = ? ? ? ?

18. Finding equations of tangents Find the equation of the normal to the curve ? = ?2 when ? = 3. Function of gradient: ?? ??= ?? Gradient when ? = 3: ? = ? Perpendicular gradient when ? = 3: ? = ? ? ? ? = 3 ?-value when ? = 3: ? = ? So equation of normal: ? ? = ? ? ?? ?

19. Second Derivative We can differentiate multiple times. For C1, you needn t understand why we might want to do so. Name Leibniz Notation Lagrange Notation ? ? ? ? ? (Original expression/function) ?? ?? ?2? ??2 ??? d?? First Derivative ? ? Second Derivative ??? ?th Derivative ?2? ??2= 30? ? = 5?3 ? ?2? ??2= 30? ? ? =

20. (questions on worksheet) Equations of tangents and normals Edexcel C1 Jan 2013 ?? ??= 2 4? 1 ? 2 ? = 6? + 3 ? ? (9, -1) Recap: If a line has gradient m and goes through ?1,?1, then it has equation: ? ?1= ? ? ?1

21. Equations of tangents and normals Edexcel C1 Jan 2012 1 2,0 ? 8,17 ? 8

22. Equations of tangents and normals Expanding gives ? = ?3+ 7?2+ 15? + 9 Thus ?? ?? is as given. ? ? = 3 52+ 14 5 + 15 = 20 So ? + 16 = 20 ? + 5 ?1= 5 + 1 5 + 32= 16 ? = 20? + 84 ? We re interested where the gradient is 20. 3?2+ 14? + 15 = 20 3?2+ 14? 5 = 0 3? 1 ? + 5 = 0 So ? = 5 (as before) or ? =1 ? 3

23. Equations of tangents and normals Edexcel C1 Jan 2011 ?? ??=3 2?2 27 1 2 8? 2 2? ? 1 243 9 4 = 32 72 + 2 + 30 = 8 ? 3 2+8 4+ 30 ?????=3 242 27 4 8 42= 24 27 1 2= 9 2 2 Thus ? =2 9 9? 4 ? ? + 8 =2 9? + 8 = 2? 8 2? 9? 16 = 0