IGCSE Math Differentiation-introduction.pptx
- 2. Starter • What do you know about Gradient?
- 3. Objectives • Define differentiation • Find the gradient function
- 4. The branch of mathematics that deals with the finding and properties of derivatives and integrals of functions • The two main types are differential calculus and integral calculus. Calculus
- 5. Differentiation, in mathematics, is the process of finding the derivative, or rate of change, of a function. In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
- 6. Can be written as when given in the form y = …. It means the derivative of y with respect to x
- 10. Activity:
- 11. Test Your Understanding 𝑦 =𝑥 7 → 𝒅𝒚 𝒅𝒙 =𝟕 𝒙 𝟔 𝑦 =3 𝑥 10 → 𝒅𝒚 𝒅𝒙 =𝟑𝟎 𝒙 𝟗 𝑓 (𝑥 )=𝑥−10 =→ 𝒅𝒚 𝒅𝒙 =−𝟏𝟎 𝒙−𝟏𝟏 𝑦 =5 𝑥 2 → 𝒅𝒚 𝒅𝒙 =𝟏𝟎 𝒙 𝟏 =𝟏𝟎 𝒙 ? ? ? ? 0 ? 1 2 3 4 5
- 12. Objective • Differentiation of multiple term
- 13. Differentiating Multiple Terms Differentiate First thing to note: If then i.e. differentiate each term individually in a sum/subtraction. 𝑑𝑦 𝑑𝑥 =2 𝑥 + 4 ? ? ? Therefore applying the usual rule: Alternatively, if you compare to , it’s clear that the gradient is fixed and . Therefore applying the usual rule: Alternatively, if you sketch , the line is horizontal, so the gradient is 0.
- 14. Activity 𝑦 =2 𝑥 2 −3 𝑥 → 𝒅𝒚 𝒅𝒙 =𝟒 𝒙 −𝟑 1 2 3 4 5 𝑦 =4 − 9 𝑥 3 → 𝒅𝒚 𝒅𝒙 =−𝟐𝟕 𝒙 𝟐 𝑦 =5 𝑥+ 1→ 𝒅𝒚 𝒅𝒙 =𝟓 𝑦 =𝑎𝑥 → 𝒅𝒚 𝒅𝒙 = 𝒂 𝑦 =6 𝑥 − 3+𝑝 𝑥 2 → 𝒅𝒚 𝒅𝒙 =𝟔+𝟐 𝒑𝒙 ? ? ? ? ? (where is a constant) (where is a constant) 6 ?
- 15. Objective • Calculate Gradient of a curve at a point
- 16. Harder Example Let a) Find the gradient of at the point b) Find the coordinates of the point on the graph of where the gradient is 8. c) Find the gradient of at the points where the curve meets the line . When Remember that the ‘gradient function’ allows you to find the gradient for a particular value of . Point is This example is important! Previously you used a value of to get the gradient . This time we’re doing the opposite: using a known gradient to get the value of . We therefore substitute for 8. a b c First find point of intersection: Solving, we obtain: or When When Once you have your , you need to work out . Ensure you use the correct equation! ? ? ?
- 17. Test Your Understanding Let a) Find the gradient of at the point b) Find the coordinates of the point on the graph of where the gradient is 5. c) Find the gradient of at the points where the curve meets the line . When Point is a b c Solving: or When When ? ? ?
- 18. Q y dy dx Gradient of curve at x a 1 7 10 a 2 2 1 3 2 x x 4 a 3 3 2 5 6 x x 0 a 4 4 2 3 x 1 a 5 2 3 4 6 x x 2 a 6 3 2 5 x 3 a 7 17 3 x 7 a 8 2 12 3 x x 10 a 9 2 5 4 x x 1 a 10 5 3 2 1 1 2 100 3 2 x x x 1 a 11 3 4x 1 2 a 12 2 3x 2 3 a 13 53 2 x 4 7 a 14 7 6 5 4 x x x x 1 a y dy dx x a 7 10 a 2 1 3 2 x x 4 a 3 2 5 6 x x 0 a 4 2 3 x 1 a 2 3 4 6 x x 2 a 3 2 5 x 3 a 17 3 x 7 a 2 12 3 x x 10 a 2 5 4 x x 1 a 5 3 2 1 1 2 100 3 2 x x x 1 a 3 4x 1 2 a 2 3x 2 3 a 53 2 x 4 7 a 7 6 5 4 x x x x 1 a y dy dx x a 7 10 a 2 1 3 2 x x 4 a 3 2 5 6 x x 0 a 4 2 3 x 1 a 2 3 4 6 x x 2 a 3 2 5 x 3 a 17 3 x 7 a 2 12 3 x x 10 a 2 5 4 x x 1 a 5 3 2 1 1 2 100 3 2 x x x 1 a 3 4x 1 2 a 2 3x 2 3 a 53 2 x 4 7 a 7 6 5 4 x x x x 1 a
- 19. Exercise 1 For each of the following, find the gradient function , and hence find the gradient of the tangent to the curve when The tangent to the curve has gradient 6. Determine the possible values of . For the curve , determine: (a) The gradient of the tangent to the curve at the point (b) The point on the curve where the gradient is 5. Find the points on the curve where the gradient is 11. 1 2 3 4 a b c d e f g ? ? ? ? ? ? ? ? ? ? ?