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GCSE AQA Further Maths Differentiation
06/04/2023
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Differentiation y=x² dy = nx² da dy You CANT do dx when... →xis in brackets →x is in denominator →x is in root form The gradient function of the curve y=f(x) is written as f'(x) or dy. olx Example f(x)=x²³ f(a+h) = (x+h)² 3 (24+h) ³-2²³ n-1 The gradient function can be used to find the gradient of the curve for any value of 2. 044 3 2²³² + 3x²³h+3₂h² +h²³ - 2²³² h h-o -- 3x² + 3₂h+h² h-x --- 3x² + 3x (0) + (0)² За Alternatively. dy da 3x² →3x2(x) =6x [cach, you) (x,y) Example f(x) = 3x² f(x+h) = 3(x+h)² 3(x+b)² - 3x² h 362₂²² +2xb+h₂²) - 3x² h 3x²³² +6xb+36²²-3x² h 6xk+ 3h² k h-so 26x +3h = 6x+3(0) =6x Differentiation Differentiation to a constant is always zero! ✓ constant 20 =0 a) Let f(x)=4x²-8x+3, Find gradient y= dy →42²³-8₂ → anything to the power of 0.1 0.8x²=8 dx •². → 8x -8 is the gradient function x = 3 :- 86)-8 =4-8 = -4 ← gradient b) Find the coordinates of point on the graph of y=f(2) where the gradient is 8. 8x-8=8 8x=16 f(x) at (²220) x = 2 Substitute → 4x²-8x +3 8.4(2) ²-8(2)+3 16-16+3 y=3 the coordinates are (2,3) d) Find the gradient of y=f(x) at the points where the Curve meets the line y=4x-5 4₂²-8x²+3=4 40-5=4 2²0 4x²_8x+3 = 4x-5 -4x+5 -4x+5 2=4 4x²-12x+8=0 x²-3x+2=0 (x-1)(x-2) = 0 2=1 8x-8 8(1)-8 8-8-0 x= 1 or x = 2 Finding gradients x=2 8x-8 8(2)-8 16-8=8
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