Relative Growth Rates of Exponential Functions

Why Compare a Rate with the Quantity Itself?

The derivative of a function gives its instantaneous rate of change. But the size of that rate alone may not tell us whether a quantity is changing quickly relative to its current size.

For example, an increase of 10 units per year is large for a quantity of 20 units and small for a quantity of 10,000 units. Dividing the rate by the current quantity measures the change relative to the amount present. For natural exponential functions, this ratio has a particularly simple form: it is constant.

In this tutorial, \(x\) denotes the input and \(f(x)=A e^{kx}\), where \(A\) and \(k\) are constants and \(e\) is the base of the natural exponential function. We will derive the ratio and interpret it in situations where \(x\) represents time or another measured input.

Deriving the Constant Ratio

The derivative rule for a natural exponential with a linear exponent gives \(\frac{d}{dx}e^{kx}=k e^{kx}\). Applying the constant multiple rule to \(f(x)=A e^{kx}\), we obtain its derivative.

If \(A\ne 0\), then \(e^{kx}>0\) for every real \(x\), so \(f(x)=A e^{kx}\ne 0\) everywhere. It is therefore valid to divide the derivative by the function:

The cancellation explains the result: the derivative and the function contain the same exponential factor and the same coefficient \(A\). The derivative has one additional factor, \(k\), so dividing by the function leaves exactly \(k\).

What the Ratio Measures

At a particular input \(x\), \(f'(x)\) is the instantaneous change in the function per unit change in the input. The quotient \(f'(x)/f(x)\) compares that change with the function’s current value. It is called the relative rate of change, or the instantaneous relative rate.

For \(f(x)=A e^{kx}\) with \(A>0\), the function is positive, so the sign of the relative rate describes its direction of change:

• If \(k>0\), then \(f'(x)=k f(x)>0\): the function is increasing, and its relative rate is \(k\).
• If \(k<0\), then \(f'(x)=k f(x)<0\): the function is decreasing, and its relative rate is \(k\). The positive quantity \(|k|\) gives the rate of relative decrease.
• If \(k=0\), then \(f(x)=A\) is constant and its derivative is zero.

When \(x\) is measured in units such as years, \(k\) has units of inverse years. The quotient \(f'(x)/f(x)\) has the same units: the units of \(f'\) are units of the quantity per year, and dividing by the quantity leaves per year.

That last wording describes an instantaneous rate, not an exact percentage change over a whole year. The derivative is local: it describes what happens at an input value, or approximately over a sufficiently small change in the input. A constant relative rate does not mean that the absolute rate \(f'(x)\) stays constant. Since \(f'(x)=k f(x)\), the absolute rate changes as the quantity changes.

Worked Examples

Why the Coefficient \(A\) Does Not Affect the Relative Rate

The constant \(A\) sets the scale of the function. Replacing \(A\) by a larger positive value multiplies every function value by that same factor, and it also multiplies every derivative value by that factor. The quotient therefore does not change.

For instance, if \(A\) is doubled while \(k\) is fixed, both \(f(x)\) and \(f'(x)\) double. Their ratio remains \(k\). Thus two positive exponential quantities can have different sizes and different absolute rates while sharing the same instantaneous relative rate.

This is useful when a rate needs to be compared across quantities of different sizes. A statement such as “the rate is 12 units per day” depends on the amount being measured. A statement such as “the instantaneous relative rate is 0.02 per day” describes how the rate compares with the current amount.

A Common Mistake: Confusing the Absolute and Relative Rates

Consider \(g(x)=7e^{3x}\). A common error is to identify \(g'(x)/g(x)\) as \(21\), because \(21\) appears when differentiating:

But \(21e^{3x}\) is the derivative, not the relative rate. The relative rate requires division by the function:

The correction is to keep the two quantities distinct: \(g'(x)=21e^{3x}\) is the absolute rate, while \(g'(x)/g(x)=3\) is the relative rate. A related error is to write \(g'(x)=3\); that drops the factor \(g(x)\). The correct relationship is \(g'(x)=3g(x)=21e^{3x}\).

Summary

For a natural exponential \(f(x)=A e^{kx}\) with \(A\ne0\), differentiation gives \(f'(x)=Ak e^{kx}\), and division by the nonzero function gives \(f'(x)/f(x)=k\). When \(A>0\), the sign of \(k\) indicates growth or decay, and its units are inverse units of the input.

The ratio is a relative rate: it compares instantaneous change with current size. It is constant for this exponential family, although the absolute rate generally varies with \(x\). Keep the distinction clear, and check the units whenever the input represents a measured quantity such as time.