Explainer: Differentiation of Exponential Functions

In this explainer, we will learn how to find the derivatives of exponential functions.

Exponential functions play an important role in mathematics and have many physical applications. For example, we use exponential functions to model the growth of bacteria, continuously compounded interest, and radioactive decay, to mention just a few of its applications. In fact, its ubiquitous use in mathematics led one mathematician to call it โ€œthe most important function in mathematics.โ€ One of the reasons why exponential functions are so important in mathematics is their properties in relation to derivatives; and in this explainer, we will explore these properties. We begin with a general definition of an exponential function.

Exponential Function

An exponential function is a function of the form ๐‘“(๐‘ฅ)=๐‘Ž๐‘,๏— where ๐‘Ž and ๐‘ are constants and ๐‘>0.

A common misconception when it comes to derivatives of exponential functions is to assume that we can apply the power rule dd๐‘ฅ๐‘ฅ=๐‘›๐‘ฅ.๏Š๏Š๏Šฑ๏Šง

This is not true as we shall see. It is important to be clear on the distinction: the power rule applies when the base of the exponent is the variable and the exponent is fixed, whereas for exponential functions the exponent is the variable and the base is fixed.

The Power RuleExponential Functions
๐‘ฅ๏Š
The variable is the base.
The exponent is constant.
๐‘๏—
The base is constant.
The variable is the exponent.

We will use the definition of the derivative to try to derive a formula for the derivative of an exponential function of the form ๐‘“(๐‘ฅ)=๐‘๏—. By definition, ๐‘“(๐‘ฅ)=๐‘“(๐‘ฅ+โ„Ž)โˆ’๐‘“(๐‘ฅ)โ„Ž;๏Ž˜๏‚โ†’๏Šฆlim hence, ๐‘“(๐‘ฅ)=๐‘โˆ’๐‘โ„Ž.๏Ž˜๏‚โ†’๏Šฆ๏—๏Šฐ๏‚๏—lim

Using the rules of exponential functions, we can rewrite this as ๐‘“(๐‘ฅ)=๐‘๐‘โˆ’๐‘โ„Ž.๏Ž˜๏‚โ†’๏Šฆ๏—๏‚๏—lim

We can now factor out ๐‘๏— as follows: ๐‘“(๐‘ฅ)=๐‘๏€พ๐‘โˆ’1โ„Ž๏Š.๏Ž˜๏‚โ†’๏Šฆ๏—๏‚lim

Since ๐‘๏— does not depend on โ„Ž, we have ๐‘“(๐‘ฅ)=๐‘๏€พ๐‘โˆ’1โ„Ž๏Š.๏Ž˜๏—๏‚โ†’๏Šฆ๏‚lim

Therefore, if the limit exists, this formula tells us that the derivative of an exponential function is proportional to the function itself.

Even more, we can say the expression lim๏‚โ†’๏Šฆ๏‚๐‘โˆ’1โ„Ž is actually the definition of the derivative of ๐‘“ at ๐‘ฅ=0; that is, ๐‘“(0)=๐‘โˆ’1โ„Ž.๏Ž˜๏‚โ†’๏Šฆ๏‚lim

Therefore, if ๐‘“ is differentiable at ๐‘ฅ=0, it is differentiable everywhere and ๐‘“(๐‘ฅ)=๐‘“(0)๐‘๏Ž˜๏Ž˜๏—.

Numerically, we can approximate the value of this limit for particular values of ๐‘. The table below includes numerical approximations of the limit.

โ„Ž2โˆ’1โ„Ž๏‚3โˆ’1โ„Ž๏‚
0.10.71773โ€ฆ1.16123โ€ฆ
0.010.69555โ€ฆ1.04466โ€ฆ
0.0010.69338โ€ฆ1.09921โ€ฆ
0.00010.69317โ€ฆ1.09867โ€ฆ

The numerical evidence certainly appears to suggest that this limit exists and, in fact, it can been proved that these limits exist.

Since the derivative of 2๏— at ๐‘ฅ=0 is less than one and the derivative of 3๏— at ๐‘ฅ=0 is greater than one, it seems reasonable to conjecture that there is a special value for 2<๐‘<3 such that its derivative at ๐‘ฅ=0 is one and, consequently, dd๐‘ฅ๐‘=๐‘๏—๏—. This value does, in fact, exist, and mathematicians denote this number ๐‘’ (Eulerโ€™s number) and it is uniquely defined as the number that satisfies lim๏‚โ†’๏Šฆ๏‚๐‘’โˆ’1โ„Ž=1.

The importance of this particular exponential function cannot be overstated. It is the unique exponential function that has a derivative of one at ๐‘ฅ=0 and its derivative is equal to its value for all ๐‘ฅ. Furthermore, we will see that all exponential functions can be expressed in terms of this exponential function. It is often referred to as the natural exponential function or simply the exponential function.

Derivative of the Exponential Function

The derivative of the exponential function is dd๐‘ฅ(๐‘’)=๐‘’.๏—๏—

This formula can be generalized to dd๐‘ฅ๏€น๐‘’๏…=๐‘˜๐‘’.๏‡๏—๏‡๏—

We will now look at some examples where we apply this formula to find the derivative of certain functions.

Example 1: Derivative of Exponential Functions

If ๐‘“(๐‘ฅ)=โˆ’5๐‘’๏Šฑ๏Šฏ๏—, find ๐‘“(๐‘ฅ)๏Ž˜.

Answer

Using the rule for differentiating exponential functions, dd๐‘ฅ๏€น๐‘’๏…=๐‘˜๐‘’,๏‡๏—๏‡๏— we have ๐‘“(๐‘ฅ)=โˆ’5(โˆ’9)๐‘’=45๐‘’.๏Ž˜๏Šฑ๏Šฏ๏—๏Šฑ๏Šฏ๏—

Example 2: Differentiating Functions Involving Exponentials

Differentiate the function ๐‘ฆ=3๐‘’โˆ’5โˆš๐‘ฅ๏—๏Žข.

Answer

Since the derivative of a sum is the sum of the derivatives, we can differentiate each term independently. Using the rule for differentiating exponential functions, we have dd๐‘ฅ3๐‘’=3๐‘’,๏—๏— and using the power rule, we have dddd๐‘ฅ๏€ฟโˆ’5โˆš๐‘ฅ๏‹=๐‘ฅ๏€ฝโˆ’5๐‘ฅ๏‰=53๐‘ฅ.๏Žข๏Ž ๏Žข๏Žฃ๏Žข๏Šฑ๏Šฑ

Therefore, combining these two expressions, we have ๐‘ฆ=3๐‘’+53๐‘ฅ.๏Ž˜๏—๏Šฑ๏Žฃ๏Žข

Example 3: Differentiating Exponential Functions

Find dd๐‘ฆ๐‘ฅ if 5๐‘ฆ๐‘’=7๐‘’๏Šจ๏—๏Šซ.

Answer

On first impression, it appears we have been given a function that is defined implicitly. However, since ๐‘’๏Šจ๏— is nonzero, we can divide both sides of the equation through by ๐‘’๏Šจ๏— which gives 5๐‘ฆ=7๐‘’๐‘’.๏Šซ๏Šจ๏—

Furthermore, we can divide through by 5 and rewrite 1๐‘’๏Šจ๏— as ๐‘’๏Šฑ๏Šจ๏—. Hence, ๐‘ฆ=7๐‘’5๐‘’,๏Šซ๏Šฑ๏Šจ๏— which is a function defined explicitly. At this point, we can differentiate the function using the rule for differentiating exponential functions, dd๐‘ฅ๏€น๐‘’๏…=๐‘˜๐‘’,๏‡๏—๏‡๏— to get dd๐‘ฆ๐‘ฅ=(โˆ’2)7๐‘’5๐‘’.๏Šซ๏Šฑ๏Šจ๏—

This can actually be expressed in terms of ๐‘ฆ as dd๐‘ฆ๐‘ฅ=โˆ’2๐‘ฆ.

Using what we know about the derivative of the natural exponential function, we can find the derivatives of general exponential functions as we will demonstrate in the next couple of examples.

Example 4: Differentiating General Exponential Functions

If ๐‘ฆ=โˆ’3ร—2๏—, determine dd๐‘ฆ๐‘ฅ.

Answer

Using the rules of exponentials and logarithms, we can write 2=๐‘’.ln๏Šจ

Raising both sides to the power of ๐‘ฅ, we have 2=๐‘’.๏—(๏Šจ)๏—ln

Hence, we can rewrite the expression for ๐‘ฆ as follows: ๐‘ฆ=โˆ’3๐‘’.(๏Šจ)๏—ln

We can now use the rule for differentiating the natural exponential function, dd๐‘ฅ๏€น๐‘’๏…=๐‘˜๐‘’,๏‡๏—๏‡๏— to get ddln๐‘ฆ๐‘ฅ=โˆ’3(2)๐‘’.(๏Šจ)๏—ln

We can convert our expression in terms of ๐‘’ back to an exponent of base 2 to get ddln๐‘ฆ๐‘ฅ=โˆ’3(2)2.๏—

The following example demonstrated the rule for differentiating general exponential functions.

Derivative of General Exponential Functions

The derivative of an exponential function of the form ๐‘๏— is ddln๐‘ฅ(๐‘)=(๐‘)๐‘.๏—๏—

This formula can be generalized to ddln๐‘ฅ๏€น๐‘๏…=๐‘˜(๐‘)๐‘.๏‡๏—๏‡๏—

Interestingly, we are now able to give an exact representation for the constants that we approximated numerically at the beginning of the explainer. In particular, ddlnddln๐‘ฅ(2)=(2)2,๐‘ฅ(3)=(3)3,๏—๏—๏—๏— which implies limlnlimln๏‚โ†’๏Šฆ๏‚๏‚โ†’๏Šฆ๏‚2โˆ’1โ„Ž=2,3โˆ’1โ„Ž=3.

Or, more generally, limln๏‚โ†’๏Šฆ๏‚๐‘โˆ’1โ„Ž=๐‘.

Example 5: Differentiating General Exponential Functions

Find the first derivative of the function ๐‘ฆ=๏€น7๏…๏Šฑ๏Šฏ๏—๏Šฑ๏Šฎ๏Šฑ๏Šจ.

Answer

Although this expression looks complex, we can use the rules of exponents to significantly simplify the expression to get something we can easily differentiate. Using the power of powers rule, (๐‘Ž)=๐‘Ž,๏Š๏‰๏Š๏‰ we can rewrite the expression for ๐‘ฆ as ๐‘ฆ=7=7.๏Šฑ๏Šจ(๏Šฑ๏Šฏ๏—๏Šฑ๏Šฎ)๏Šง๏Šฎ๏—๏Šฐ๏Šง๏Šฌ

At this point, we might wonder how we can handle the fact that we have a binomial in the exponent. However, we can use the multiplicative rule of exponents, ๐‘Ž๐‘Ž=๐‘Ž,๏Š๏‰๏Š๏Šฐ๏‰ to rewrite this as ๐‘ฆ=๏€น7๏…7.๏Šง๏Šฌ๏Šง๏Šฎ๏—

We now have an expression we can easily differentiate. Using the rule for differentiating general exponential functions, ddln๐‘ฅ๏€น๐‘๏…=๐‘˜(๐‘)๐‘,๏‡๏—๏‡๏— we have ddln๐‘ฆ๐‘ฅ=๏€น7๏…(187)7.๏Šง๏Šฌ๏Šง๏Šฎ๏—

We can recombine the product of exponential functions to rewrite this as ddln๐‘ฆ๐‘ฅ=18๏€น7๏…7.๏Šง๏Šฎ๏—๏Šฐ๏Šง๏Šฌ

In our final example, we will consider a word problem involving differentiating exponential functions.

Example 6: Derivatives of Exponential Functions in a Real-World Context

The radioactive decay of radon-222 is modeled by the following formula: ๐‘(๐‘ก)=๐‘๏€ผ12๏ˆ,๏Šฆ๏‘‰๏‘‰๏Ž ๏Žก where ๐‘(๐‘ก) is the remaining quantity, in grams, of radon-222 that has not decayed after ๐‘ก days, ๐‘๏Šฆ is the initial quantity of radon-222, and ๐‘ก๏Ž ๏Žก is its half-life.

A particular sample initially contained 10 grams of radon-222. Given that the half-life of radon-222 is ๐‘ก=3.8215๏Ž ๏Žกdays, find the rate of decay of the sample 10 days later. Give your answer to three significant figures.

Answer

When we are asked to find the rate of something, we should immediately think derivative. We have been asked for the rate of decay; therefore, we need to differentiate the function ๐‘(๐‘ก)=๐‘๏€ผ12๏ˆ๏Šฆ๏‘‰๏‘‰๏Ž ๏Žก with respect to ๐‘ก. Using the rule for differentiating general logarithms, ddln๐‘ฅ๏€น๐‘๏…=๐‘˜(๐‘)๐‘,๏‡๏—๏‡๏— we have ddln๐‘๐‘ก=๐‘๐‘ก๏€ผ12๏ˆ๏€ผ12๏ˆ.๏Šฆ๏Ž ๏Žก๏‘‰๏‘‰๏Ž ๏Žก

Using the laws of logarithms, we can rewrite this as ddln๐‘๐‘ก=โˆ’๐‘2๐‘ก๏€ผ12๏ˆ.๏Šฆ๏Ž ๏Žก๏‘‰๏‘‰๏Ž ๏Žก

We need the rate of decay when ๐‘ก=10; therefore, substituting this in, along with ๐‘ก=3.8215๏Ž ๏Žก and ๐‘=10๏Šฆ, we have ddlngramsperday(3signi๏ฌcant๏ฌgures)๐‘๐‘ก|||=โˆ’1023.8215๏€ผ12๏ˆ=0.29570โ€ฆ=0.296.๏๏Šฒ๏Šง๏Šฆ๏Ž ๏ŽŸ๏Žข๏’๏Žง๏Žก๏Ž ๏Žค

Key Points

  • The derivative of the exponential function is dd๐‘ฅ(๐‘’)=๐‘’,๏—๏— which can be generalized to dd๐‘ฅ๏€น๐‘’๏…=๐‘˜๐‘’.๏‡๏—๏‡๏—
  • Similarly, the derivative of a general exponential function of the form ๐‘๏— is ddln๐‘ฅ(๐‘)=(๐‘)๐‘.๏—๏— This formula can also be generalized to ddln๐‘ฅ๏€น๐‘๏…=๐‘˜(๐‘)๐‘.๏‡๏—๏‡๏—
  • Before differentiating, it can be useful to simplify or manipulate the expression using the rules of exponents.
  • Remember not to confuse the power rule and the rule for differentiating exponential functions.
    The Power RuleDerivatives of Exponential Functions
    ๐‘ฅ๏Š
    The variable is the base.
    The exponent is constant.
    dd๐‘ฅ(๐‘ฅ)=๐‘›๐‘ฅ๏Š๏Š๏Šฑ๏Šง
    ๐‘๏—
    The base is constant.
    The variable is the exponent.
    ddln๐‘ฅ(๐‘)=(๐‘)๐‘๏—๏—

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