The video explains that the derivative of an exponential function \(a^{t}\) is proportional to the function itself. By examining the difference quotient \(\frac{a^{t+dt}-a^{t}}{dt}\) and using the property \(a^{t+dt}=a^{t}a^{dt}\), the limit as \(dt\to0\) yields a constant that depends only on the base \(a\). This constant is the natural logarithm of \(a\):
\[
\frac{d}{dt}a^{t}=a^{t}\ln(a).
\]
The special base \(e\approx2.718\) is defined by \(\ln(e)=1\), so \(\frac{d}{dt}e^{t}=e^{t}\); its slope equals its height at every point. Using the chain rule, \(\frac{d}{dt}e^{kt}=ke^{kt}\). Any exponential \(a^{t}\) can be rewritten as \(e^{\ln(a)\,t}\), making the proportionality constant \(\ln(a)\) explicit. This form is useful because in many natural processes (population growth, cooling, interest) the rate of change is proportional to the quantity itself, and the constant in the exponent \(e^{kt}\) directly represents that proportionality rate.
1. The derivative of 2^t with respect to t equals 2^t multiplied by ln(2) ≈ 0.6931.
2. For any positive base a, the derivative of a^t equals a^t multiplied by ln(a).
3. The base e ≈ 2.71828 is defined such that the derivative of e^t equals e^t.
4. Any exponential a^t can be rewritten as e^{ln(a)·t}.
5. Using the chain rule, the derivative of e^{kt} equals k·e^{kt}.
6. For example, the derivative of e^{3t} equals 3·e^{3t}.
7. The proportionality constant ln(a) is obtained by evaluating the limit (a^{dt}−1)/dt as dt→0.
8. Numerically, ln(2) ≈ 0.6931, ln(3) ≈ 1.0986, and ln(8) ≈ 2.079.
9. In natural processes where a quantity’s rate of change is proportional to the quantity itself, the quantity follows an exponential function.
10. Writing an exponential as e^{kt} makes the constant k directly interpretable as that proportionality constant.
11. The number e is not fundamental to the function; it is a convenient base that gives the exponent a clear meaning.
12. The derivative of a^t is proportional to a^t, with the proportionality factor being the natural logarithm of the base.