Rate of Change of Quantity

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If a quantity y varies with respect to another quantity 'x' satisfying some rule y = f(x), in other words if y is a function x, then

represents the rate of change of y with respect to x

For x = x₀,

dy/dx at x₀ is called the rate of change of y with respect to x at x₀.

If y is a function of t

x is a function of t

That is if x = f (t) and y = g(t)

We know that

The rate of change of y with respect to x in this case, can be found by finding out the rate of change of y with respect to t and that of x with respect to t.

Example:

A balloon which always remain spherical is being inflated by pumping in 900 cubic centimetres of gas per second. Final the rate at which the radius of the balloon increasing when the radius is 15cm.

Let r and V be the variables for radius and volume of the balloon respectively.

We have to find

For a sphere

Differentiating both sides with respect to t, we have