If a quantity y depends on and varies with a quantity x, then the rate of change of y with respect to x isÂ **dy/dx**.

Let y=f(x) be a function of x. LetÂ be the change in y corresponding to a small changeÂ . ThenÂ represents the change in y due to a unit change in x. In other words,Â represents the average rate of change of y w.r.t. x as x changes from x toÂ

AsÂ the limiting value of this average rate of change ofÂ y with respect to x in intervalÂ becomes the instantaneous rate of change of y w.r.t. x

ThusÂ = instantaneous rate of change of y w.r.t. x

= rate of change of y w.r.t. xÂ

The word ”instantaneous” is often dropped .

HenceÂ represents the rate of change of y w.r.t. x for a definite value of x

### Example

**Find the rate of change of the area of a circle with respect to its radiusÂ **

**Solution**

Let A be the area of the circle. Then

Thus, the rate of changeÂ of the area of the circle w.r.t. its radius r is