Differentiation From First Principles

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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, (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 dx ("delta" x) and the vertical distance between A and B is called dy, the coordinates of B are (x + dx, y + dy).
From the coordinate geometry section, we know that the gradient of a straight line joining two points is:
y₂ - y₁, where the two points are (x₁, y₁) and (x₂, y₂)
x₂ - x₁
In this case, the two points are (x, y) and (x + dx, y + dy). So substituting these values into the formula, the gradient of the chord is:
y + dy - y  =  dy     (pronounced "delta y by delta x")
x + dx - x      dx

This is the gradient of the chord. The gradient of the curve is the gradient of the chord when the chord has no length- i.e. when it is a tangent. This will happen when dx = 0 .
The gradient of the curve is therefore:
 lim       ( dy )
dx®0    ( dx )
This basically means that the gradient is dy/dx as dx approaches or "tends to" (®) zero.

We can rewrite the coordinates of (x, y) as (x, f(x)) and the coordinates of (x + dx, y + dy) as (x + dx, f(x + dx)), since y is a function of x (y = f(x)).
So the gradient of the curve is:
 lim      (y + dy - y)
dx®0   (x + dx - x)

since y = f(x) and y + dy = f(x + dx):
Gradient is:
 lim      f(x + dx) - f(x)
dx®0           dx

This is denoted by dy/dx ("dee y by dee x"). dy/dx is known as the derivative of y with respect to x.

So, in summary,
dy   =     lim      f(x + dx) - f(x)
dx        dx®0           dx

Example

Find the formula for the gradient of the graph y = x² .
dy  =    lim      (x + dx)² - x²
dx      dx®0           dx

=   lim     x² + 2xdx + (dx)² - x²
  dx®0             dx
=   lim      2xdx + (dx)²
   dx®0           dx
The dx on the denominator cancels with those on the numerator.

Therefore dy/dx =
 lim      2x + dx
dx®0
When dx becomes zero, dy/dx = 2x.

Therefore the gradient of y = x² is 2x.
For example, at the point (2, 4), the gradient is 2x = 4 .

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