Derivatives of exponential functions: WACE Year 12 Mathematics Methods Unit 3

What this dot point is asking

SCSA Unit 3 singles out the natural exponential function $e^{x}$ because of one remarkable property: its gradient at every point equals its height. This dot point asks you to know that property, derive the chain-rule extension, and differentiate exponentials confidently in both the calculator-free and calculator-assumed sections of the WACE written examination.

Why $e^{x}$ is its own derivative

Consider $f(x)=a^{x}$. From first principles,

The factor $a^{x}$ comes out because it does not depend on $h$. The remaining limit is a constant that depends only on the base $a$; it is the gradient of $a^{x}$ at $x=0$. The number $e\approx 2.71828$ is defined precisely as the base for which that limiting constant is exactly $1$. Therefore

This makes $e^{x}$ the natural model for any process whose rate of growth is proportional to its current size.

The chain rule with exponentials

Whenever the exponent is more than a single $x$, the chain rule supplies a factor equal to the derivative of the exponent. For $y=e^{kx}$ the exponent is $kx$, whose derivative is $k$, giving $y'=k\,e^{kx}$. For a general inner function $f(x)$ the factor is $f'(x)$.

General bases

For a base other than $e$, write $a^{x}=e^{x\ln a}$ and apply the chain rule. The exponent $x\ln a$ has derivative $\ln a$, so

For example $\dfrac{d}{dx}2^{x}=2^{x}\ln 2$. When $a=e$ the factor $\ln e=1$ recovers $\dfrac{d}{dx}e^{x}=e^{x}$.

Exponential models and proportional growth

The property $\dfrac{d}{dx}e^{kx}=k\,e^{kx}$ makes the exponential the natural model for any quantity whose rate of change is proportional to its current size. If $Q=Q_{0}e^{kt}$, then

so the growth rate is always $k$ times the current amount. A positive $k$ gives growth (populations, compound interest); a negative $k$ gives decay (radioactive material, cooling, drug concentration). Recognising $\dfrac{dQ}{dt}=kQ$ as the signature of exponential change is a recurring SCSA theme, and the constant $k$ is read straight off the exponent.

Reading the gradient

Because $e^{x}>0$ for all $x$, the curve $y=e^{x}$ is always increasing and always concave up: both $y'$ and $y''$ equal $e^{x}$, which is positive everywhere. This is why the graph has no stationary points and a horizontal asymptote $y=0$ as $x\to-\infty$. The same positivity means $y=e^{kx}$ never has a stationary point either: $k e^{kx}=0$ has no solution, so exponential curves are strictly monotonic.