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Derivera Polynom

Derivera Polynom. We have looked many times at the graph f(x)=x 2 .

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Derivera Polynom

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  1. Derivera Polynom Wehavelookedmanytimes at the graph f(x)=x2. A tangent is a linewhich touches a curve at onepointonly. In the graph f(x) = x2 you cansee a tangent at the point (1;1). You coulddraw a tangent at (2;4) to find the gradient where x=2. Note that f(2) = x2 = 4 The gradient at anypoint on f(x) is given by the gradient (lutning) of the tangent to that point.

  2. Derivera Polynom The gradient (lutning) at anypoint on f(x) is given by the gradient (lutning) of the tangent to that point. Wedo not need to draw the tangent ourselvesifweknow f(x) in order to find the gradient (lutning).

  3. Derivera Polynom Ifwedraw tangents at (1;1), (2;4); (3;9) and (4;16) trust mewhen I tell you that you willdiscover the resultsbelow!

  4. Derivera Polynom Can you see a patternbetween x and f’(x) for f(x)=x2?

  5. Derivera Polynom If f(x) = x2 then f’(x) = 2x. NOTE: WE will PROVE this later in the term as the syllabussayswe must do. If f(x) = x3 then f’(x) = 3x2 If f(x) = x4 – 2x2then f’(x) = 4x3 – 4x Can you see the patternyet? (Remember that 4x=4x1)

  6. Derivera Polynom Here is the rule! If f(x) = axn f’(x) = naxn-1 Example f(x) = 3x6 a=3, n=6 so f’(x) = 3.6x6-1  f’(x) = 18x5 Think of the rule as 'multiply by the index, then reduce the index by 1'.

  7. Try These For the followingfunctions f(x), find f’(x) f(x) = x2+7x f(x) = r2 f(x) = 16x3 – 4x2 + 2x d) f(x) = 3x(4x + 2) Think of the rule as 'multiply by the index, then reduce the index by 1'.

  8. Answers f’(x) = 2x+7 f’(x) = 2 r f’(x) = 48x2 – 8x + 2 f’(x) = 24x + 6 Think of the rule as 'multiply by the index, then reduce the index by 1'.

  9. OtherImportantFacts What is f’(x) if f(x) = x3 + x2 – 10? Write f(x) as x3 + x2 – 10x0 since 100 = 1 f’(x) = 3x2 + 2x NOTE: f’(k) = 0 where k is anynumber, for example, 25.

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