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Differentiation

Differentiation helps us find the rate of change in the relationships between two variables. As we briefly saw in the Introduction to Algebra, speed measures the change in the distance travelled and the time taken. Also the two variables here, viz. distance travelled and time taken are related to each other.

Using Linear Geometry, let’s look at a function y = mx + b, where m is the gradient or slope of the line.

Gradient of the line m is represented by the change in ‘y’ values to the change in ‘x’ values.

m  =   {change in value of y}/{change in value of x}

     =   {Delta y}/{Delta x}

 {Delta y} and  {Delta x} represent small changes in the variables y and x respectively.

When  {Delta x} right 0 (very small change), we define

 lim{Delta x right 0}~{{Delta y}/{Delta x}}  =   {dy}/{dx} .

This represents differentiation or a derivative of a real function y with respect to x. Let’s look at a few examples to show this representation of change in 2 variables.

 

Example: Here we will look at a real life situation where we’ll use differentiation to find the rate of change.

A water tank has a height of 120m, and its area of cross-section is 2000m2. Water is released from the tank and the water level drops by 1 metre every minute.

Let the height of the water level be h. After t minutes, the height of the water in the tank is given by –

h = 120 – 1 x t  =  120 – t

Rate at which the water level drops in the tank is given by  {dh}/{dt}  = -1 m/minute.

Volume of the water in the tank after t minutes will be v = A x h  =  2000 (120 – t)  =  240000 – 2000t

Rate at which the volume drops in the tank is given by  {dv}/{dt}  = -2000 m3 (which is 2000 m2 x -1 m/min).

Notice that in the equation h = 120 – t, the gradient of the line function is -1, i.e. the rate of change of flow.

Similarly in the equation V = 240000 – 2000t, the gradient of the function V is -2000, which is the rate of change of volume of water in the tank i.e. the rate at which water flows out of the tank, and it is 2000 m3

  1.  Dependent variable is Height (H)

    Independent variable is Time (T)

    H and T are like y and x respectively in the above illustration.

    Rate of change in height =   {dH}/{dT}

  2.  Dependent variable is Profit,

    Independent variable is Price

    Rate of change in profit with respect to price =   {dProfit}/{dPrice}

  3.  Dependent variable is distance

    Independent variable is time

    Rate of change  =   {ds}/{dt} , where s is the distance and t is time.

    This rate of change of distance with respect to time is called speed or velocity.