From: 3blue1brown
This article explores the derivatives of exponential functions, highlighting the unique properties of Euler’s number e and its significance in calculus [00:00:26].
Derivatives of Exponential Functions
To understand the derivative of exponential functions, consider a function like 2t [00:00:32]. If we model this as the total mass of a population doubling every day [00:00:55], the rate of growth over a full day (e.g., between day 3 and day 4) equals the population size at the start of that day [00:02:17]. For example, between day 3 and 4, it grows from 8 to 16, a rate of 8 new creature masses per day, which is the initial population size [00:01:46]. This might suggest the derivative of 2t is itself [00:02:25].
However, for a true derivative, we must consider increasingly smaller changes in time (dt) [00:02:48]. The change in the function per unit time is expressed as (2t+dt - 2t) / dt [00:03:19]. A core property of exponentials allows us to rewrite 2t+dt as 2t * 2dt [00:04:14]. This allows factoring out 2t, resulting in 2t * (2dt - 1) / dt [00:04:40].
As dt approaches 0, the term (2dt - 1) / dt approaches a constant value, approximately 0.6931 [00:05:32]. This indicates that the derivative of 2t is 2t multiplied by this constant [00:05:52]. Similarly, for a base of 3t, the constant is about 1.0986 [00:06:45], and for 8t, it’s around 2.079 [00:07:08]. These constants are not random but follow a pattern [00:07:22].
The Special Nature of Euler’s Number
A key question arises: Is there a base for which this proportionality constant is exactly 1? [00:07:42] The answer is Euler’s number e, approximately 2.71828 [00:07:55]. This is the defining property of e [00:08:04]. The derivative of et is et itself [00:08:30].
Graphically, this means that for any point on the graph of et, the slope of the tangent line at that point is equal to the height of the point above the horizontal axis [00:08:35].
Connecting Bases: The Role of Natural Logarithm
This special property of e helps explain the mystery constants for other bases using the chain rule [00:08:52] [00:09:01]. The derivative of ekt is k * ekt [00:09:42].
Any number, such as 2, can be written as eln(2) [00:09:56]. Thus, the function 2t is equivalent to e(ln(2) * t) [00:10:10]. Applying the chain rule, the derivative of this function is proportional to itself, with the proportionality constant being the natural logarithm of 2 [00:10:24]. When calculated, ln(2) is approximately 0.6931, which matches the constant found earlier for 2t [00:10:38].
In general, the proportionality constant for the derivative of a base a to the power t is simply ln(a) [00:10:46].
Why e is Preferred in Calculus Applications
In calculus applications, exponential functions are almost always written as ekt, rather than some base to the power of t [00:11:08]. This is because any exponential function like 2t or 3t can be rewritten in this form [00:11:14].
The advantage of using e is that the constant k in the exponent carries a meaningful interpretation [00:11:45]. In natural phenomena where a variable’s rate of change is proportional to itself (e.g., population growth, cooling, compound interest) [00:11:56], the function describing that variable over time will be an exponential [00:12:45]. When expressed as ekt, the constant k directly represents the proportionality constant between the size of the changing variable and its rate of change [00:13:04].