Derivatives of Exponential Functions

derivative of exponentialUnderstanding derivatives can be an arduous task due to the many rules they have for various types of equations. Finding the derivative of a polynomial is nowhere near the same as finding the derivative to a trigonometric function. There’s also the importance of understanding the quotient rule and product rule as well.

One concept that is a must is learning how to take the derivatives of logs and the derivative of exponents. Exponential derivatives are actually a simple concept to grasp, and with time, you can easily master it.

As a part of Calculus, learning about derivatives is important to understanding most of it. There’s a wide variety of derivatives out there, and they all have different rules that you have to follow. The Udemy course Calculus I – Applications of Derivatives is the first place you should go if you want to learn how to use all of the various types of derivatives that are out there.

If the concept is still new to you, you can brush up on your knowledge of Calculus, not just with derivatives, but with all of it by checking out the Udemy course AP Calculus AB that doesn’t just help you with Calculus, but also help prepares you for the Advanced Placement test you take at the end of the course.

Different Types of Exponential Functions

The derivative of exponents requires understanding two important types of exponents. The first type of exponent is one where the base is e. In mathematics, e is a very specific number that is commonly used with exponents.

In fact, e is such a specific number that if there is any exponent that has the base of e and you’re asked to take the derivative of it, the answer is the same as the question.

For example, the derivative of e5 is e5. Even if the question were to ask you what the derivative of e286 the answer to the equation would still be the same as the question, or more specifically in this case, e286.

The other cases where you have to take the derivative of an exponent is when the base is anything else.

Proving the Derivative of Exponential Functions 

In mathematics, to properly explain something, people use proofs to help others understand why something is the way it is. There are several proofs that come along with exponents and all regard some of the more special cases that exponents are involved in as opposed to other derivative functions.

In most situations, the derivative of an exponent is similar to when the base is e.

If the base of the exponent is any other number, then the proof goes as follows.

First of all, you must remember that the derivative of the natural log of x, also known as the derivative of ln (x), is 1/x. Then you will do the proof as follows.

Assume that y = ax.

If you take the natural log of each side, the problem would look like ln y = ln ax. Use the power rule and bring the x down to the right side of the natural log of a in order to make the equation represent ln y = x ln a.

Take the derivative of both sides, which would make y 1/y and the x ln a = ln a.

Finally, bring y over to the side and substitute it for ax like you did before. This turns the final product into ax ln a. 

Examples of Taking the Derivative of an Exponential Function 

As you’ve seen above, exponential functions don’t require a lot of mathematics to fully understand, but instead it requires you to understand the rules of various forms of derivatives, such as the derivative of logarithms.

An example of an exponential derivative problem would be to take the derivative of 25. Simply put, you would multiply the entire equation by its natural log and get ln (25). You would then pull the 5 down to the left side of the ln and get the answer 5 ln(2), and that would be it.

If you’ve already gotten the hang of this, you should check out the Calculus 1 – Applications of Derivatives course on Udemy to learn more about the various types of derivative equations you will have to do. There’s also the Calculus I Essentials course that gives you a rundown of all the rules you need to know in order to succeed in your calculus course.