y_{2} - y_{1}, where the two points are (x_{1}, y_{1}) and (x_{2}, y_{2})

x_{2} - x_{1}

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

## Comments

No comments have yet been made