A football is kicked into a pool. It creates a circular ripple that traveles outward with a speed of 2 ft/sec. How fast does the circular area enlarge after 3 seconds?

If the radius of the circular ripple is r and the area of the circular ripple is s. The area of the circular ripple is

     s = πr²

The enlarging circular area with respect to time t is s^'_t or ds/dt

     ds/dt = d(πr²)/dt

             = d(π(r²))/dt

             = 2πr(dr/dt)

When t = 3 seconds,

     r = (2)(3) = 6

So

     ds/dt = 2πr(dr/dt)

         = 2π(6)(2)

         = 24π ft²/sec

In order to evaluate the rate of the enlarging circular area, the concept of derivative has been employed. Derivative is widely used in science and engineering research. For example, how to calculate the linear density of a non-homogeneous rod. The mass from the left end to the point x is

     mass = f(x)

The mass between x₂ and x₁ is

     Δmass = f(x₂) - f(x₁)

So the average density of the rod between x₂ and x₁ is

     average density = Δmass/Δx = ( f(x₂) - f(x₁))/(x₂-x₁)

Letting x₂ approaches x₁, the linear density ρ of the rod at point x₁ is the limit of the average density when Δx approaches 0.

In other words, the linear density of the rod is the derivative of the mass with respect to length.

Since the concept of derivative can apply to many problems in science and engineering, it is important for students to understand its physical meaning.