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
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