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Join The Student Room TodayBe part of the UK's largest and fastest growing student community. It's free to join and a lot of fun - Get inspired, express your ideas, interact and share Revision:Differentiation from 1st PrinciplesFrom The Student RoomTSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Differentiation from 1st Principles NB: many syllabuses do not require a knowledge of this - check yours! Gradient of a curve A curve does not have a constant gradient. At any point on a curve, the gradient is equal to the gradient of the tangent at that point (a tangent to a curve is a line touching the curve at one point only). For example, the gradient of the below curve at A is equal to the gradient of the tangent at A, which is XY. An approximation to the gradient at any point can be found by drawing a chord. A chord joins together two points on a curve. The closer together these two points are, the closer one gets to the actual gradient of the graph at the point in question. Therefore in the above diagram, AB and AC are chords. The gradient at A is closer to the gradient of AC than AB, since the chord AC is shorter. Every time one makes the chord shorter, the gradient of the chord gets closer and closer to the gradient of the curve at A. Eventually, when the chord becomes so short that it is a tangent, the gradient of the graph will equal the gradient of this tangent. The derivative We can use algebra to find out what the gradient of this tangent will be. A is any point, From the coordinate geometry section, we know that the gradient of a straight line joining two points is: This is the gradient of the chord. The gradient of the curve is the limit of the gradient of the chord as the chord length gets smaller - i.e. when The gradient of the curve is therefore: If we are given that y = f(x), we can rewrite the coordinates as
and the gradient of the curve is
This is denoted So, in summary,
Find the formula for the gradient of the graph Choose two points
We know that i.e. Therefore the gradient of For example, at the point (5, 25), the gradient is 2x = 2*5 = 10. Comments |
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