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

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

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

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

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

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

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

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

If y = xn = nxn-1 Lets do some differentiating dy dx The general rule (very important) is :- If y = xn “Times by the power and reduce the power by 1” dy dx = nxn-1 E.g. if y = x2 = 2x dy dx E.g. if y = x3 = 3x2 dy dx E.g. if y = 5x4 = 5 x 4x3 = 20x3 dy dx

You just substitute the x value in Example 1 E.g. if y = x3 + 13x = 3x2 +13 dy dx You can just add them together So the gradient at x=3 is ….. dy dx = 3 x 32 +13 = 27 + 13 = 40 You just substitute the x value in

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

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

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