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Core 3 Differentiation Learning Objectives: Review understanding of differentiation from Core 1 and 2 Review understanding of differentiation from Core 1 and 2 Understand how to differentiate e x Understand how to differentiate e x Understand how to differentiate ln ax Understand how to differentiate ln ax

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Differentiation means…… Differentiation means…… Finding the gradient function. Finding the gradient function. The gradient function is used to calculate the gradient of a curve for any given value of x, so at any point. The gradient function is used to calculate the gradient of a curve for any given value of x, so at any point. Differentiation Review

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The Key Bit The general rule (very important) is :- If y = x n dy dx = nx n-1 E.g. if y = x 2 = 2x dy dx E.g. if y = x 3 = 3x 2 dy dx E.g. if y = 5x 4 = 5 x 4x 3 = 20x 3 dy dx dy dx

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A differentiating Problem The gradient of y = ax 3 + 4x 2 – 12x is 2 when x=1 What is a? dy dx = 3ax 2 + 8x -12 When x=1 dy dx = 3a + 8 – 12 = 2 3a - 4 = 2 3a = 6 a = 2

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Finding Stationary Points At a maximum At a minimum dy dx =0 dy dx =0 + dy dx > dy dx < 0 - d2yd2y dx 2 < 0 d2yd2y dx 2 > 0

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Differentiation of a x Compare the graph of y = a x with the graph of its gradient function. Adjust the values of a until the graphs coincide.graphs coincide

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Differentiation of a x Summary The curve y = a x and its gradient function coincide when a = The number 2.718….. is called e, and is a very important number in calculus See page 88 and 89 A1 and A2

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Differentiation of e x

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The gradient function f(x )and the original function f(x) are identical, therefore The gradient function f(x )and the original function f(x) are identical, therefore The gradient function of e x is e x The gradient function of e x is e x i.e. the derivative of e x is e x i.e. the derivative of e x is e x If f(x) = e x f `(x) = e x Also, if f(x) = ae x f `(x) = ae x

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Differentiation of e x Turn to page 90 and work through Exercise A Turn to page 90 and work through Exercise A

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Derivative of ln x ln x is the inverse of e x ln x is the inverse of e x The graph of y=ln x is a reflection of The graph of y=ln x is a reflection of y = e x in the line y = x This helps us to differentiate ln x This helps us to differentiate ln x If y = ln x then x = e y so If y = ln x then x = e y so So Derivative of ln x is = 1

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Differentiation of ln x Live page

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Differentiation of ln 3x Live page

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Differentiation of ln 17x Live page

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Summary - ln ax (1) f(x) = ln x f(1) = 1 the gradient at x=1 is 1 f(4) = 0.25 the gradient at x=4 is 0.25 f(x) = ln 3x f(x) = ln 17x f(1) = 1 the gradient at x=1 is 1 f(4) = 0.25 the gradient at x=4 is 0.25 f(1) = 1 the gradient at x=1 is 1 f(4) = 0.25 the gradient at x=4 is 0.25

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Summary - ln ax (2) For f(x) = ln x For f(x) = ln ax Whatever value a takes…… the gradient function is the same f(1) = 1 the gradient at x=1 is 1 f(4) = 0.25 the gradient at x=4 is 0.25 f(100) = 0.01 f(0.2) = 5 the gradient at x=100 is 0.01 the gradient at x=0.2 is 5 The gradient is always the reciprocal of x For f(x) = ln x For f(x) = ln ax f `(x) = 1/x

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Examples If f(x) = ln x If f(x) = ln 7x f `(x) = 1/x If f(x) = ln x 3 If f(x) = ln 11x 3 f(x) = ln + ln x 3 f(x) = ln 11 + ln x 3 Dont know about ln ax 3 f(x) = ln + 3 ln x f(x) = ln ln x f `(x) = 3 (1/x) f `(x) = 3/x Constants go in differentiation

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If y = x n dy dx = nx n-1 if f(x) = ae x f `(x) = ae x if g(x) = ln axg`(x) = 1/x Summary if h(x) = ln ax n h`(x) = n/x h(x) = ln a + n ln x

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Differentiation of e x and ln x Classwork / Homework Classwork / Homework Turn to page 92 Turn to page 92 Exercise B Exercise B Q1,3, 5 Q1,3, 5

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