C1 Chapter 7 Differentiation Dr J Frost Last modified: 13 th October 2013

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.

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

Suppose we’re finding the gradient of y = x 2 at the point A(3, 9). Gradient = 6 ? How could we find the gradient?

For the curve y = x 2, we find the gradient for these various points. Can you spot the pattern? Point(1, 1)(4, 16)(2.5, 6.25)(10, 100) Gradient28520 For y = x 2, gradient = 2x ? Let’s prove it...

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

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, 0 0, inf/inf, inf – inf, 1 inf, inf^0, and 0 x inf

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

Your Turn: What is the gradient of y = x 3 ? (x, x 3 ) (x + h, (x+h) 3 ) δxδx δyδy ? ? ?

Differentiating x n Can you spot the pattern? yx2x2 x3x3 x4x4 x5x5 x6x6 dy/dx2x3x 2 4x 3 5x 4 6x 5 If y = x n, then dy/dx = nx n-1  ? If y = ax n, then dy/dx = anx n-1 In general, scaling y also scales the gradient ? y = x 7  dy/dx = 7x 6 y = x 10  dy/dx = 10x 9 y = 2x 3  dy/dx = 6x 2 f(x) = 2x 3  f’(x) = 6x 2 f(x) = x 2 + 5x 4  f’(x) = 2x + 20x 3 y = 3x 1/2  dy/dx = 3/2 x -1/2 ? ? ? ? ? ?

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

Differentiating cx and c ? ? ? ?

Exercises FunctionGradientPoint(s) of interestGradient at this point dy/dx = 10x 4 (2, 64)160 f’(x) = 21x 2 (3, 189)189 dy/dx = 2 + 2x(4, 24)10 f’(x) = 3x 2 – x -2 (2, 11.5)11.75 f’(x) = 4x 3 (2, 16)32 dy/dx = 3x 2 (3, 31), (-3, -23)27 3 Be sure to use the correct notation for the gradient. ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Test your knowledge so far... Edexcel C1 May 2012 ? ?

Turning more complex expressions into polynomials ? ? ? ? ? ? ?

Exercise 7E (Page 115) 1 2 ? ? ? ? ? ? ? ? ?

Finding equations of tangents ? ? ? ?

? ?

Second Derivative We can differentiate multiple times. For C1, you needn’t understand why we might want to do so. NameLeibniz NotationLagrange Notation (Original expression/function) First Derivative Second Derivative ? ?

Equations of tangents and normals Edexcel C1 Jan 2013 (9, -1) ? ? ? (questions on worksheet)

Equations of tangents and normals Edexcel C1 Jan 2012 ? ?

Equations of tangents and normals ? ? ?

Edexcel C1 Jan 2011 ? ? ?