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Revision:Differentiation from 1st Principles

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TSR 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.

image:diff1.gif

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.

image:diff2.gif

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, (x,y). To find the gradient at A, we need to find the gradient of the tangent at A. Let B be a point which is just a little further along the graph. The gradient of the chord AB is approximately the gradient of A. If the horizontal distance between A and B is called \delta x (read: "delta x") and the vertical distance between A and B is called \delta y, the coordinates of B are (x+\delta x,y+\delta y).

From the coordinate geometry section, we know that the gradient of a straight line joining two points is: \dfrac{y_2-y_1}{x_2-x_1} . In this case, the two points are (x,y) and (x+\delta x,y+\delta y). So substituting these values into the formula, the gradient of the chord is \dfrac{\delta y}{\delta x}

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 \delta x becomes so small that the chord is a tangent.

The gradient of the curve is therefore: \displaystyle \lim_{\delta x \rightarrow 0} \left( \frac{\delta y}{\delta x} \right), i.e. the value that \dfrac{\delta y}{\delta x} tends to as \delta x tends to 0.

If we are given that y = f(x), we can rewrite the coordinates as (x, f(x)) and (x+\delta x,f(x+\delta x)). So the gradient of the chord is

\displaystyle \frac{\delta y}{\delta x} = \frac{f(x+\delta x) - f(x)}{x + \delta x - x}

and the gradient of the curve is

\displaystyle \lim_{\delta x \rightarrow 0} \frac{\delta y}{\delta x} = \lim_{\delta x \rightarrow 0} \frac{f(x+\delta x) - f(x)}{\delta x}.

This is denoted \dfrac{\text{d} y}{\text{d} x} or f^\prime (x).

So, in summary, \displaystyle \frac{\text{d} y}{\text{d} x} = f^\prime (x) = \lim_{\delta x \rightarrow 0} \frac{f(x+\delta x) - f(x)}{\delta x}.


Example:

Find the formula for the gradient of the graph y=x^2.

Choose two points (x, x^2),\; (x+\delta x, (x+\delta x)^2). The gradient between them is:

\displaystyle \frac{\delta y}{\delta x} = \frac{(x+\delta x)^2 - x^2}{x + \delta x - x}

\displaystyle = \frac{2x \delta x + (\delta x)^2}{\delta x}

= 2x +\delta x.

We know that \displaystyle \frac{\text{d} y}{\text{d} x} = \lim_{\delta x\rightarrow 0} (2x +\delta x)

i.e. \displaystyle \frac{\text{d} y}{\text{d} x} = 2x.

Therefore the gradient of y=x^2 is \displaystyle \frac{\text{d} y}{\text{d} x} = 2x.

For example, at the point (5, 25), the gradient is 2x = 2*5 = 10.

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