# 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:

More generally if u is any differentiable function, then:

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:

Here are some direct, computational examples.

1.

2.

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4.

5.

### Generalization to base a

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

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:

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

Here are some examples applying the logarithmic derivative formula.

Using the Derivative of the Natural Logarithm

1.

2.

3.

4.

5.

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

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:

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

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:

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

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:

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

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

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:

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

Then if we differentiate both sides, we arrive at:

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

Note the similarity between the derivatives of:

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:

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

Differentiating both sides of this, we have:

or

So for example if:

then:

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

and then apply the Chain Rule to obtain:

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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