Tangents and Differentiation The gradient at any point on any curve is defined as the GRADIENT FUNCTION How do we find the gradient function of a curve?

Repeat for smaller values of 𝑥 Curves do not have a constant rate of change, i.e. the gradient is not constant. How do we calculate the rate of change at any point on a curve? This is defined to be the gradient of the tangent drawn at that point. 𝑓(𝑥) 𝑓(𝑎) 𝑎 Repeat for smaller values of 𝑥 𝑥 𝑓 𝑥 = 𝑥 2 𝑎𝑛𝑑 𝑎=1 x 2 𝑚 𝑠𝑒𝑐 3

This shows a sequence of points 𝑄 1 , 𝑄 2 … This shows a sequence of points 𝑄 1 , 𝑄 2 …. getting closer and closer to the point P. What does this show? 𝑓 𝑥 = 𝑥 2 𝑎𝑛𝑑 𝑎=1 x 2 1.5 1.1 1.001 𝑚 𝑠𝑒𝑐 3 2.5 2.1 2.001

𝑓 𝑥 = 𝑥 2 𝑎𝑛𝑑 𝑎=1 As we choose values of 𝑥 getting closer to 𝑎=1, we see that the lines from P to each of the Q’s get nearer and nearer to becoming a tangent at P. If we let the point Q approach the point P along the curve, then the gradient of the line should approach the gradient of the tangent at P, and hence the gradient of the curve. 𝑥 𝒎 𝒔𝒆𝒄 2 3 1.5 2.5 1.1 2.1 1.001 2.001 1.00001 2.00001 The values of 𝒎 𝒔𝒆𝒄 appear to be getting closer to 2. In the case we write: lim 𝑥 →1 𝒎 𝒔𝒆𝒄 =2

Algebraic Method Let Q be the point 1+ℎ, 1+ℎ 2 Gradient of PQ = (1+ℎ) 2 −1 1+ℎ −1 = 1+2ℎ+ ℎ 2 −1 ℎ = ℎ( 2+ℎ) ℎ = 2+ℎ As Q approaches closer and closer to P, ℎ→0 𝑠𝑜, 2+ℎ→2 This agrees with the previous method. This finds the gradient at one point of curve, but how can we find the gradient at any point?

Algebraic Method To find the gradient at 𝑃 𝑥, 𝑥 2 , take a point Q 𝑥+ℎ, 𝑥+ℎ 2 𝑥+ℎ, 𝑥+ℎ 2 on the curve close to P. Gradient of PQ = (𝑥+ℎ) 2 − 𝑥 2 𝑥+ℎ −𝑥 = 𝑥 2 +2𝑥ℎ+ ℎ 2 − 𝑥 2 ℎ = ℎ(2𝑥+ℎ) ℎ = 2𝑥+ℎ As Q approaches closer and closer to P, ℎ→0 𝑠𝑜 2𝑥+ℎ→2𝑥 At any point on 𝑦= 𝑥 2 the gradient of the curve is 2𝑥. So for 𝑦= 𝑥 2 the gradient function is 2𝑥 𝑦= 𝑥 2 (𝑥+ℎ) 2 (𝑥+ℎ) 2 − 𝑥 2 𝑥 2

Differentiation – First Principles Differentiation is the process of finding the gradient function of a curve. For any curve 𝑦=𝑓(𝑥), the gradient can be defined as follows: Let 𝑃(𝑥,𝑦) be any point on the curve and Q 𝑥+𝛿𝑥, 𝑦+𝛿𝑦 be a point on the curve near P. 𝛿𝑥 and 𝛿𝑦 are the small changes in 𝑥 and 𝑦 between P and 𝑄. Gradient of P𝑄 = 𝑦+𝛿𝑦 −𝑦 𝑥+𝛿𝑥 − 𝑥 = 𝛿𝑦 𝛿𝑥 Gradient at P = lim 𝛿𝑥→0 𝛿𝑦 𝛿𝑥 = 𝑑𝑦 𝑑𝑥 ( 𝑑𝑦 𝑑𝑥 is the notation used for the gradient function or derivative of a curve)

lim 𝛿𝑥→0 𝛿𝑦 𝛿𝑥 = 𝑑𝑦 𝑑𝑥 = 𝑓′(𝑥) = lim 𝛿𝑥→0 𝑓 𝑥+𝛿𝑥 −𝑓(𝑥) 𝛿𝑥 Definition The derivative of a function 𝑓 with respect to a point 𝑥 is defined by: lim 𝛿𝑥→0 𝛿𝑦 𝛿𝑥 = 𝑑𝑦 𝑑𝑥 = 𝑓′(𝑥) = lim 𝛿𝑥→0 𝑓 𝑥+𝛿𝑥 −𝑓(𝑥) 𝛿𝑥 Find the derivative of 𝑓 𝑥 =𝑥 3 𝑓 ′ 𝑥 = lim 𝛿𝑥→0 𝑓 𝑥+𝛿𝑥 −𝑓(𝑥) 𝛿𝑥 = lim 𝛿𝑥→0 (𝑥+𝛿𝑥) 3 − 𝑥 3 𝛿𝑥 = lim 𝛿𝑥→0 𝑥 3 +3 𝑥 2 𝛿𝑥 +3𝑥 (𝛿𝑥) 2 + (𝛿𝑥) 3 − 𝑥 3 𝛿𝑥 = lim 𝛿𝑥→0 3 𝑥 2 +3𝑥 𝛿𝑥 + (𝛿𝑥) 2 = 3 𝑥 2 The derivative of 𝒙 𝟑 is 𝟑 𝒙 𝟐