Using the Derivatives of Natural Base e & Logarithms

Instructor: Christopher Haines
In this lesson, we learn the derivative formulas for exponential and logarithmic functions. In addition, we learn how to apply them through several concrete examples.

Exponential Function Derivative

The derivative of the exponential function of natural base e is the following:

Derivative of e^x

More generally if u is any differentiable function, then:

generalized exponential function derivative

This extension version of the exponential derivative follows directly from the Chain Rule. As a refresher, the Chain Rule is stated here.

Chain Rule Theorem

Suppose f and g are two differentiable functions. Define h = (f o g)(x). Then:

Chain Rule Formula

Here are some direct, computational examples.

Using the Derivative Formula for Exponential Functions

1.

constant multiple function for u(x)

2.

Exponential function with u being quadratic

3.

trig function as u

4.

miscellaneous exponential derivative

5.

ProductRuleExponentialPolynomial

Generalization to base a

In general if a > 0, then the derivative formula is:

Base_a_exponentialderivativeformula

The next section covers the derivative of the natural logarithmic function and its generalization to all positive bases. The natural logarithmic function is the inverse function of the exponential function, base e. Similarly, there is a logarithm function base a which corresponds (uniquely) with the exponential function base a. (a > 0)

Logarithmic Function Derivative

The derivative of the natural logarithmic function is the following:

Logarithmic Derivative (u(x) = x)

More generally if u is a differentiable function, it follows from the Chain Rule that:

General Derivative formula for logarithm (base e)

Here are some examples applying the logarithmic derivative formula.

Using the Derivative of the Natural Logarithm

1.

Example 1 Natural Log

2.

Example 2 Log derivative

3.

Example 3 Natural Log Derivative

4.

Example 4 Natural Log Derivative

5.

Log Derivative Example 5

If a is any base, (positive number) then by using the change of base formula, it is easily seen that:

Derivative Formula for log base a

Thus, the generalized formula for both the exponential and logarithmic functions is simply a multiple times the natural case of base e. In the next section, we give a proof for the formulas in the case of a = e. (natural logarithm) The formulas for the general case of a > 0 are a simple modification in the proof of a = e, and therefore, we leave it to the reader to validate those other two formulas.

Proof of Exponential and Logarithmic Derivatives (Base e)

Proof of Logarithmic and Exponential Derivative Formula (base e)

In this section, we give a proof of the logarithmic and exponential derivatives. We start by defining the number e as:

Definition of e

This is just one of many ways of defining e. Next using the definition of derivative, we have:

First line of proof

Now note that as h tends to zero, the fraction x/h tends to infinity. (assuming x > 0, which would only make sense in the domain of the logarithm) Further by rewriting the last expression as:

Second line of proof

and then passing the limit to the inside of natural logarithm, (permitted since the natural logarithm is a continuous function) we can conclude that:

Third line of proof

So that proves the general natural logarithm derivative formula if we combine the previous result with the Chain Rule. Next, we prove the exponential derivative formula. We begin by first noting:

Fourth line of proof

Next, we can use the logarithm derivative formula to observe that:

Fifth line of proof

By combining the last two equations, it becomes apparent that:

Sixth line of proof

So, that completes the proof of the natural logarithmic and natural exponential derivative formulas. (with the help of the Chain Rule)

In the upcoming section, we continue the topic of exponential and logarithmic derivatives to discuss logarithmic differentiation.

Logarithmic Differentiation

Suppose our function is:

x to the power of x

If we take the natural logarithm of both sides of the equation, we have:

log of both sides

Then if we differentiate both sides, we arrive at:

Derivative of both sides

Finally multiplying both sides by y, and recalling how y is defined, we obtain:

Solve for y solution

Note the similarity between the derivatives of:

Comparison of exponential functions

Both of them have the original function multiplied by a logarithm. However, the difference in the latter is there is an extra term.

More generally, suppose our function is of the form:

General Exponential Form

explanation of u and v

Then by taking the natural logarithm of both sides, we have:

natural log step

Differentiating both sides of this, we have:

derivative of logs of both sides

or

solution after differentiating

So for example if:

Example on log differentiation

then:

Derivative of example given

Alternatively, we can compute the derivative of the general function as follows. First write the original as:

alternate exponential form

and then apply the Chain Rule to obtain:

Alternate method of obtaining formula

This is precisely the same expression we obtained using the method of logarithmic differentiation. This should not be a surprise though, since there is a relationship between the derivatives of a function and its inverse. (assuming all necessary conditions hold)

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