Derivatives of Exponential and Logarithmic Functions

1. The Star of the Show: The Natural Exponential Function, $e^x$

Before we can differentiate $e^x$, it's helpful to understand what the number $e$ actually is and why it's so special. It's not just a random constant; it arises naturally in many areas, including finance, population growth, and, as we'll see, calculus itself.

To get some intuition for this, let's explore the origin of $e$ through the concept of continuous compounding.

As the video shows, $e$ is the limit of $(1+1/n{)}^n$ as $n$ approaches infinity, approximately equal to 2.718. This 'natural' base has a unique and beautiful property when it comes to differentiation.

Let's see this property derived from first principles. The next part of the same video will show what happens when we try to differentiate a general exponential function $a^x$, and why choosing the base $e$ is so convenient.

This leads us to one of the most elegant results in all of calculus:

The derivative of $e^x$ is simply $e^x$.
$\frac{d}{dx}{e}^x=e^x$
This means the value of the function $e^x$ at any point is equal to the gradient (slope) of its tangent line at that same point. It is its own derivative.

Of course, we'll often combine this with the Chain Rule. For a differentiable function $u=f(x)$:
$\frac{d}{dx}{e}^u=e^u\cdot \frac{du}{dx}$
Example: Find the derivative of $y=e^{x^2+3x}$.

• The outer function is $e^u$, and the inner function is $u=x^2+3x$.
• The derivative of the inner function is $\frac{du}{dx}=2x+3$.
• Applying the Chain Rule: $\frac{dy}{dx}=e^{x^2+3x}\cdot (2x+3)$.

2. The Inverse: The Natural Logarithmic Function, $\ln (x)$

Every exponential function has an inverse: a logarithmic function. The inverse of the natural exponential function $y=e^x$ is the natural logarithm, written as $y={\log }_e(x)$ or, more commonly, $y=\ln (x)$.

Let's continue with the video to see this relationship and then derive the derivative of $\ln (x)$.

The derivation reveals another simple and powerful rule:

The derivative of $\ln (x)$ is $\frac{1}{x}$.
$\frac{d}{dx}\ln (x)=\frac{1}{x}$

When combined with the Chain Rule for a differentiable function $u=f(x)$:
$\frac{d}{dx}\ln (u)=\frac{1}{u}\cdot \frac{du}{dx}=\frac{u^{\prime }}{u}$
This pattern—the derivative of the inside function over the original inside function—is extremely common.

3. General Bases: $a^x$ and ${\log }_{a}x$

What if the base is not $e$? How do we find the derivative of $2^x$ or ${\log }_{10}(x)$? We don't need to go back to first principles. Instead, we can use a clever trick to convert any base into base $e$, allowing us to use the rules we just learned.

The key is to remember that $a=e^{\ln a}$. This allows us to rewrite any exponential or logarithmic function in terms of its natural counterpart.

To see how this is done and to get the final two rules, please read the following text.

Let's summarise the results from that reading:

1. Derivative of $a^x$:
By rewriting $a^x=(e^{\ln a}{)}^x=e^{x\ln a}$ and using the Chain Rule, we get:
$\frac{d}{dx}{a}^x=\frac{d}{dx}{e}^{x\ln a}=e^{x\ln a}\cdot (\ln a)=a^x\ln a$
Rule: $\frac{d}{dx}{a}^x=a^x\ln a$

2. Derivative of ${\log }_{a}x$:
By using the change of base formula ${\log }_{a}x=\frac{\ln x}{\ln a}$, we get:
$\frac{d}{dx}{\log }_{a}x=\frac{d}{dx}\left(\frac{\ln x}{\ln a}\right)=\frac{1}{\ln a}\cdot \frac{d}{dx}(\ln x)=\frac{1}{\ln a}\cdot \frac{1}{x}$
Rule: $\frac{d}{dx}{\log }_{a}x=\frac{1}{x\ln a}$

4. Summary of Rules and Practice

We now have our four key rules for this family of functions. This summary image is a great reference.

The final step is to practice applying these rules in combination with the Product, Quotient, and Chain rules, just as we did with trigonometric functions.

The same web resource we used before has a fantastic collection of worked examples. Studying them is an efficient way to see these rules in action.

Conclusion

Congratulations on completing your core toolkit for differentiation! You can now find the derivative of nearly any elementary function you're likely to encounter by combining the rules you've learned.

Key Takeaways:

• The natural exponential function $e^x$ is its own derivative: $\frac{d}{dx}{e}^x=e^x$.
• The derivative of the natural logarithm $\ln (x)$ is $\frac{1}{x}$.
• For a general base $a$, the derivatives are:
  • $\frac{d}{dx}{a}^x=a^x\ln a$
  • $\frac{d}{dx}{\log }_{a}x=\frac{1}{x\ln a}$
• You can derive the general base rules by converting to base $e$ and using the Chain Rule.
• Always be on the lookout for opportunities to use logarithm properties to simplify an expression before you differentiate.

In our next lesson, we will learn a new technique called implicit differentiation. This will allow us to find derivatives of equations that aren't written as explicit functions of $x$, such as the equation of a circle $x^2+y^2=1$.