Exponential functions are powerful tools in calculus, modeling growth and decay in nature and finance. They have unique properties, like $a^0 = 1$ and $a^{x+y} = a^x \cdot a^y$, which make them versatile for real-world applications.

The natural exponential function, $e^x$, is special because its derivative equals itself. This property simplifies calculations in many fields, from population dynamics to compound interest, making it a cornerstone of differential calculus.

Exponential Functions and Their Derivatives

Properties of exponential functions

• Exponential functions have the general form $f(x) = a^x$, where $a$ is a positive constant not equal to 1
  • When $a > 1$, the function increases as $x$ increases (growth)
  • When $0 < a < 1$, the function decreases as $x$ increases (decay)
• Key properties of exponential functions include:
  • $a^0 = 1$ for any positive value of $a$
  • $a^{x+y} = a^x \cdot a^y$ demonstrates the product rule for exponents
  • $(a^x)^y = a^{xy}$ shows how to handle an exponent raised to another exponent
  • $\frac{a^x}{a^y} = a^{x-y}$ illustrates the quotient rule for exponents
• The natural exponential function, denoted by $e^x$, has a base of $e$, which is approximately equal to 2.71828
  • The constant $e$ arises naturally in many mathematical and scientific contexts (growth rates, compound interest)

Applications of exponential derivatives

• The derivative of an exponential function $f(x) = a^x$ is given by the formula:
  • $\frac{d}{dx}a^x = a^x \ln(a)$, where $\ln(a)$ represents the natural logarithm of $a$
• For the natural exponential function $f(x) = e^x$, the derivative simplifies to:
  • $\frac{d}{dx}e^x = e^x$, since $\ln(e) = 1$
• When finding the derivative of a composite function like $f(x) = e^{g(x)}$, apply the chain rule:
  • $\frac{d}{dx}e^{g(x)} = e^{g(x)} \cdot g'(x)$, where $g'(x)$ is the derivative of the inner function $g(x)$
  • Example: $\frac{d}{dx}e^{x^2+1} = e^{x^2+1} \cdot (2x)$

Natural vs general exponential derivatives

• The derivative of the natural exponential function $f(x) = e^x$ is equal to itself:
  • $\frac{d}{dx}e^x = e^x$, making it a unique and important function in calculus
• In contrast, the derivative of a general exponential function $f(x) = a^x$ includes the natural logarithm of the base $a$:
  • $\frac{d}{dx}a^x = a^x \ln(a)$, requiring an extra step to evaluate the derivative
• When solving problems involving exponential derivatives, carefully identify whether the function is a natural exponential ($e^x$) or a general exponential ($a^x$) to apply the correct derivative formula
  • Example: $\frac{d}{dx}2^x = 2^x \ln(2)$, while $\frac{d}{dx}e^x = e^x$

Real-world exponential derivative problems

• Exponential growth and decay problems frequently appear in various fields:
  • Population growth: $P(t) = P_0 e^{kt}$, where $P_0$ is the initial population and $k$ is the growth rate constant
    • Example: If a population starts with 100 individuals and grows at a rate of 5% per year, the population after 10 years is $P(10) = 100e^{0.05 \cdot 10} \approx 164$
  • Radioactive decay: $A(t) = A_0 e^{-\lambda t}$, where $A_0$ is the initial amount and $\lambda$ is the decay constant
    • Example: If a radioactive substance has a half-life of 10 years, the decay constant is $\lambda = \frac{\ln(2)}{10} \approx 0.0693$
• Continuously compounded interest is another application of exponential functions and their derivatives:
  • The balance $A(t)$ after time $t$ with an initial principal $P$, annual interest rate $r$, and continuous compounding is given by:
    1. $A(t) = Pe^{rt}$
  • To find the instantaneous rate of change of the balance at any time $t$, take the derivative: 2. $\frac{d}{dt}A(t) = Pre^{rt}$
  • Example: If you invest $1000 at a 6% annual interest rate with continuous compounding, after 5 years, the balance will be $A(5) = 1000e^{0.06 \cdot 5} \approx 1349.86$, and the instantaneous rate of change at that time is $\frac{d}{dt}A(5) = 1000 \cdot 0.06 \cdot e^{0.06 \cdot 5} \approx 80.99$