If derivative dsdt is written, then it is meant that the rate of change of distances with respect to the time t. When one quantity y varies with other quantity x, then dydx or (f’(x)) denotes the rate of change of y with respect to x. 

If two variables x and y are varying with respect to another variable t then, if x=f(t) and y=g(t), then by chain rule, 

                         dydx = dydt/ dxdt, if dxdt ≠ 0

The derivative as the rate of change of function has many applications which include acceleration in physics, marginal function in economics, and population growth rates also. 

DEFINE RATE OF CHANGE-

Let us take two variables x and y. If x and y are varying with respect to another variable t, i.e., if x=f(t)and y=g(t). Using the chain rule, 

                     dydx = dydt/ dxdt, if dxdt ≠ 0

Hence, the rate of change of y with respect to x can be given by the rate of change of y and that of x both with respect to t. 

dydx is positive if y increases as x increases and will be negative if y decreases as x increases. 

To explain this, let us understand it with an example-

Find the rate of change of the area of the circle per second with respect to its radius r when r=10cm. 

Area A of the circle with radius r is given by A= r2. 

The rate of change of area A with respect to radius r = dAdr= ddr(r2)=2Πr.

r= 10 cm , dAdr=20

Hence, the  Area of the circle is changing at the rate of 20 cm2/s. 

RATE OF CHANGE IN DIFFERENTIATION –

We will use differentiation to know whether a function is increasing or decreasing. Let I be an interval contained in the domain of real value function f. Then f is said to be 

(i) increasing on I if x1<x2 in I which implies f(x1) < f(x2) for all x1, x2 I.  

(ii) decreasing on I, if x1,x2 in I which implies f(x1) < f(x2) for all x1, x2 I.

(iii) constant on I, if f(x)=c for all x I, where c is constant. 

(iv) decreasing on I if  x1<x2 in I which implies f(x1) ≥ f(x2) for all x1, x2 I.

(v) strictly decreasing on I if   x1<x2 in I which implies f(x1) > f(x2) for all x1, x2 I.

The increasing or decreasing of a function can be defined in the other way also. Let x0 be any point in the domain of real value function f. Then, f is said to be increasing, decreasing at x0 if there exists an open interval I which contains x0 such that f is increasing, decreasing in I. 

Let f be the continuous on [a, b] and differentiable on the open interval (a, b). Then, 

(i) f is increasing in [a, b] if f'(x) > 0 for each x (a, b)

(ii) f is decreasing in [a, b] if f'(x)<  0   for each x (a, b)

(iii) f is constant function in [a, b] if  f'(x)= 0   for each x (a, b)

RATE OF CHANGE EXAMPLE-

The examples related to the rate of change of differentiation is given below- 

1.The volume of a cube is increasing at the rate of  9 cubic centimeters per second. How fast is the surface area increasing when the length of an edge is 20 centimeters? 

 Solution- Let x be the length of a side. 

                           V be the volume 

                           S be the surface area of the cube

                     Then,  V=x3 and S=6×2, where x is a function of t. 

                                dNdt= 9cm2/ s  (Given)

                      Therefore,  9= dNdt = ddt(x2 ). dxdt (By chain rule)

                                                         = 3×2 dxdt 

                                 Or          dxdt = 3×2

                        Now,      dSdt= ddt( 6×2) =ddt(6×2). dxdt = 12x(3×2) =36x

                        When x=10 cm, dSdt= 3.6 cm2/sec 

2. Show the function f given by f(x)=x3-3×2+4x, x Ris increasing on R.

           Solution-       f'(x)= 3×2-6x +4

                                             = 3( x2- 2x+1)+1  

                                             = 3(x-1)2+1 >0, in every interval on R.

                                   Therefore, the function f is increasing on R.  

CONCLUSION-

The rate of change of a function is the rate of change of one variable with respect to another variable. It is one of the important applications of derivatives. Above we have discussed how to identify the function if it is increasing, decreasing, or constant. We know that rate of change of function dydx is positive if y increases as x increases and will be negative if y decreases as x increases.