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Core 3 Differentiation

Core 3 Differentiation. Learning Objectives: Review understanding of differentiation from Core 1 and 2 Understand how to differentiate e x Understand how to differentiate ln a x. Differentiation Review. Differentiation means…… Finding the gradient function.

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Core 3 Differentiation

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

  2. Differentiation Review • Differentiation means…… • 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.

  3. The Key Bit dy dx dy dx dy dx dy dx = nxn-1 dy dx E.g. if y = 5x4 = 5 x 4x3 = 20x3 E.g. if y = x2 = 2x E.g. if y = x3 = 3x2 The general rule (very important) is :- If y = xn

  4. dy dx dy dx = 3ax2 + 8x -12 = 3a + 8 – 12 = 2 When x=1 A differentiating Problem The gradient of y = ax3 + 4x2 – 12x is 2 when x=1 What is a? 3a - 4 = 2 3a = 6 a = 2

  5. At a minimum At a maximum dy dx dy dx dy dx dy dx < 0 d2y d2y dx2 dx2 + - - + =0 =0 < 0 > 0 > 0 Finding Stationary Points

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

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

  8. Differentiation of ex

  9. Differentiation of ex • The gradient function f’(x )and the original function f(x) are identical, therefore • The gradient function of ex is ex • i.e. the derivative of ex is ex f `(x) = ex If f(x) = ex f `(x) = aex Also, if f(x) = aex

  10. Differentiation of ex • Turn to page 90 and work through Exercise A

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

  12. Differentiation of ln x Live page

  13. Differentiation of ln 3x Live page

  14. Differentiation of ln 17x Live page

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

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

  17. Examples f `(x) = 1/x If f(x) = ln 7x If f(x) = ln 11x3 Don’t know about ln ax3 f(x) = ln 11+ln x3 f(x) = ln 11+3ln x f `(x) = 3(1/x) Constants go in differentiation f `(x) = 3/x

  18. = nxn-1 dy dx Summary If y = xn f `(x) = aex if f(x) = aex if g(x) = ln ax g`(x) = 1/x if h(x) = ln axn h`(x) = n/x h(x) = ln a +nlnx

  19. Differentiation of ex and ln x • Classwork / Homework • Turn to page 92 • Exercise B • Q1 ,3, 5

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