## Derivative As An Instantaneous Rate Of Change

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# Derivative As An Instantaneous Rate Of Change

We now give the interpretation of the derivative as a rate of change which, in turn, has applications to business and economics.

Let us consider a function y = f (x). We use the symbol Δx to denote the change in the variable x. For example, if x change from 2 to 4, then the change in x is Δx = 4 – 2 = 2. Δx may be positive or negative. Similarly, we use the symbol Δx to denote the change in the variable y. Usually, Δy depends on the values of x and Δx . According to the equation y = f (x), at any point x, the corresponding value of y is given by y = f (x). Suppose that there is a change, Δx in x, so that the new value of x is x+ Δx . Consequently, the new value of y is f (x+ Δx).

Thus, the change, Δy, in y is given by

Δy = f (x+ Δx – f (x).

The ratio Δy/Δx is called the average rate of change of y with respect to x over the interval from x to x + Δy

If Δx were to become smaller and smaller, the average rate of change over the interval from x to x + Δx, would be close to what we might call the instantaneous rate of change of y with respect to x. We define the limit of the average rate of change of the function y = f (x) as Δx → 0 to be the instantaneous or point or simply the rate of change of y with respect to x. Thus the instantaneous rate of change of a function y = f (x) with respect to x is defined to be

lim Δy/Δx = lim f (x+Δx) - f (x)

Δx→0 Δx→0 Δx

The last limit is just the definition of the derivative of f with respect to x. Thus

the instantaneous rate of change of y with respect to x = dv/dx

= lim Δy/Δx

From this, it follows that if Δx is close to 0, then Δy/Δx is close to dy/dx. In other words,

Δy/Δx ≈ dy/dx

Therefore,

Δy ≈ dy/dx Δx.

Thus if x changes by Δx, then the change in y, Δy, is approximately dy/dx times the change in x. In particular, if x changes by 1, an estimate in the change in y is dy/dx.

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Let us consider a function y = f (x). We use the symbol Δx to denote the change in the variable x. For example, if x change from 2 to 4, then the change in x is Δx = 4 – 2 = 2. Δx may be positive or negative. Similarly, we use the symbol Δx to denote the change in the variable y. Usually, Δy depends on the values of x and Δx . According to the equation y = f (x), at any point x, the corresponding value of y is given by y = f (x). Suppose that there is a change, Δx in x, so that the new value of x is x+ Δx . Consequently, the new value of y is f (x+ Δx).

Thus, the change, Δy, in y is given by

Δy = f (x+ Δx – f (x).

The ratio Δy/Δx is called the average rate of change of y with respect to x over the interval from x to x + Δy

If Δx were to become smaller and smaller, the average rate of change over the interval from x to x + Δx, would be close to what we might call the instantaneous rate of change of y with respect to x. We define the limit of the average rate of change of the function y = f (x) as Δx → 0 to be the instantaneous or point or simply the rate of change of y with respect to x. Thus the instantaneous rate of change of a function y = f (x) with respect to x is defined to be

lim Δy/Δx = lim f (x+Δx) - f (x)

Δx→0 Δx→0 Δx

The last limit is just the definition of the derivative of f with respect to x. Thus

the instantaneous rate of change of y with respect to x = dv/dx

= lim Δy/Δx

From this, it follows that if Δx is close to 0, then Δy/Δx is close to dy/dx. In other words,

Δy/Δx ≈ dy/dx

Therefore,

Δy ≈ dy/dx Δx.

Thus if x changes by Δx, then the change in y, Δy, is approximately dy/dx times the change in x. In particular, if x changes by 1, an estimate in the change in y is dy/dx.

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