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Rate of change / Differentiation (2)

Rate of change / Differentiation (2). Gradient of curves Differentiating. dy dx. dy dx. Recall: A bit of new symbology. y. x. = “difference in y” “difference in x”. = gradient of line. PRONOUNCED “dee-why by dee-ex”. Gradient of Curves. Zoom. (3.1,3.1 2 ) B.

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Rate of change / Differentiation (2)

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  1. Rate of change / Differentiation (2) Gradient of curves Differentiating

  2. dy dx dy dx Recall: A bit of new symbology y x = “difference in y” “difference in x” = gradient of line PRONOUNCED “dee-why by dee-ex”

  3. Gradient of Curves Zoom (3.1,3.12) B Gradient = “difference in y” “difference in x” = 9.61 - 9 3.1 - 3 = 6.1 A (3,9) The tangent to the curve gives the gradient at that point y=x2 y (3,9) x (3.1,9.61)

  4. Gradient of Curves Zoom (3.01,3.012) B Gradient = “difference in y” “difference in x” = 9.0601 - 9 3.01 - 3 = 6.01 A (3,9) The tangent to the curve gives the gradient at that point y=x2 y (3,9) x (3.01,9.0601)

  5. As the interval in x decreases it tends to a definite value - always twice ‘x’

  6. A bit of theory dy dx x (delta x) is the difference in the x coordinates y y Gradient = x x As x gets smaller, it gives the gradient of the tangent

  7. More Terminology is the symbol used for the gradient of the curve dy dx dy dx The process of finding is called differentiating The gradient function is known as the derivative dy dx

  8. s s s t t t ds dt ds dt ds dt Graphs of displacement and gradient vs time The curves of gradient are always one power less (in x) than the original curves “y=ax3+bx2+cx+d” “y=ax2+bx+c” “y=mx+c” t t t “y=const.” “y=ax2+bx+c” “y=mx+c”

  9. Lets do some differentiating 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 “Times by the power and reduce the power by 1”

  10. dy dx dy dx So the gradient at x=3 is ….. = 3 x32 +13 = 27 + 13 = 40 E.g. if y = x3 + 13x = 3x2 +13 You just substitute the x value in Example 1 You can just add them together

  11. dy dx dy dx dy dx dy dx dy dx dy dx dy dx Find for these functions :- Gradient at x=2 = 6x = 6x5 = 25x4 = 120x9 = 3x2 + 2x = 18x2 + 6x + 11 = 12 = 192 = 400 = = 12 +4 = 16 =72+12+11 =95 y= 3x2 y = x6 y = 5x5 y = 12x10 y = x3 + x2 y = 6x3 + 3x2 + 11x

  12. dy dx Harder Examples y = 3x(2x2 +9) Expand bracket first E.g. if y = 3x(2x2 +9) y = 6x3 +27x = 18x2 +27 Divide through Express as fractional or negative indices The rules still work

  13. Harder Examples - your turn dy dx y = 2x2(3x3 +x) Expand bracket first E.g. if y = 6x5 +2x3 = 30x4 +6x2 Divide through Express as fractional or negative indices The rules still work

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