MCV4U Derivatives of Exponential Functions: Notes and Practice

A colony of bacteria grows faster when there are more bacteria. Money earning compound interest grows faster when the balance is bigger. A hot drink cools faster when it is much hotter than the room. In each case the RATE of change depends on the AMOUNT (for the drink, the amount by which it is hotter than the room). Exponential functions are the functions whose rate of change is proportional to the amount, and in this lesson you will see it appear directly in their derivatives.

Two agar plates covered with hundreds of small cream and blue-green bacterial colonies, with two empty clear dishes below them. — PhotoEach dot on these plates is a colony of E. coli bacteria. A growing colony adds cells faster the more cells it already has, which is the rate-proportional-to-amount pattern behind exponential functions.PlaxcoLab, CC BY-SA 2.0, via Wikimedia Commons

The slope of aˣ is a multiple of aˣ

\frac{a ^{x + h} - a ^{x}}{h} = \frac{a ^{x} a ^{h} - a ^{x}}{h} = a^{x} \cdot \frac{a ^{h} - 1}{h} ⟶ a^{x} \cdot k, where k = h \to 0 lim \frac{a ^{h} - 1}{h}

Factor out a to the x. What is left does not depend on x at all, so the derivative is the function times a constant k.

So $\frac{d}{d x} a^{x} = k a^{x}$ : the derivative is a vertical stretch of the function itself, and the ratio $\frac{f ^{'} ( x )}{f ( x )}$ is the same constant $k$ at every $x$ . Setting $x = 0$ shows that $k$ is simply the slope of the tangent at $x = 0$ , where $a^{0} = 1$ . The table estimates $k$ numerically for three bases.

Base a	Estimate of k using h = 0.001	Exact value of k
$2$	$0.69339$	$ln 2 \approx 0.69315$
$e \approx 2.71828$	$1.00050$	$ln e = 1$
$3$	$1.09922$	$ln 3 \approx 1.09861$

Estimates are (a to the h, minus 1) divided by h, rounded to 5 decimal places. The exact values are explained further down the page.

Fig 1y = 2ˣ, eˣ and 3ˣ near x = 0, with tangent slopes at x = 0 of ln 2 ≈ 0.693, 1 and ln 3 ≈ 1.099. Only base e has slope equal to height everywhere.

For base $2$ the constant is a bit less than $1$ , and for base $3$ it is a bit more. Somewhere between $2$ and $3$ there is a base whose constant is exactly $1$ . That base is the number $e \approx 2.71828$ , an irrational number like $π$ . For $y = e^{x}$ , the slope of the tangent at every point is equal to the height of the graph at that point:

\frac{d}{d x} (e^{x}) = e^{x}

The function e to the x is its own derivative.

Video3Blue1Brown shows why the derivative of an exponential function is the function times a constant, and why base e is the one where that constant is exactly 1, the same argument this section builds.Watch on YouTube

The natural logarithm undoes eˣ

The NATURAL LOGARITHM, $ln x$ , is $lo g_{e} x$ : the exponent you must put on $e$ to get $x$ . It is the inverse of $e^{x}$ , so each one reverses the other: $e^{l n x} = x$ for $x > 0$ , and $ln (e^{x}) = x$ for every $x$ . As with any pair of inverse functions, their graphs are reflections of each other in the line $y = x$ , and every point $(a, b)$ on one graph appears as $(b, a)$ on the other.

Fig 2y = eˣ and y = ln x reflected in the line y = x, with the matching pairs (0, 1) and (1, 0), and (1, e) and (e, 1), marked.

Now the mystery constant can be named. Because $e^{l n a} = a$ , any base can be written as a power of $e$ : $a^{x} = (e^{l n a})^{x} = e^{x l n a}$ . The chain rule, with inner function $x ln a$ whose derivative is the constant $ln a$ , gives $\frac{d}{d x} a^{x} = e^{x l n a} \cdot ln a = a^{x} ln a .$ So $k = ln a$ , which matches the table: $ln 2 \approx 0.693$ and $ln 3 \approx 1.099$ .

\frac{d}{d x} (a^{x}) = a^{x} ln a \frac{d}{d x} e^{g (x)} = e^{g (x)} g^{'} (x)

The general exponential rule (a greater than 0, a not equal to 1), and the chain rule with e: copy the exponential exactly, then multiply by the derivative of the exponent.

Loading interactive…

Choose eˣ or 2ˣ, drag a point along the curve, and compare the tangent slope with the height of the graph.

Power function or exponential function?

x^{2}

the variable is the base and the exponent is fixed: use the power rule, giving

2 x

. In

2^{x}

the base is fixed and the variable is the exponent: use the exponential rule, giving

2^{x} ln 2

. Using the power rule on

e^{x}

to get

x e^{x - 1}

is one of the most common errors in this unit.

Looking ahead

The reflection also shows how

ln x

behaves. Reflecting a tangent line in

y = x

swaps rise and run, so slopes at matching points are reciprocals. At

(1, e)

the slope of

e^{x}

e

, so at

(e, 1)

the slope of

ln x

\frac{1}{e}

. In general the slope of

ln x

at any

x > 0

\frac{1}{x}

, a result you will use in later calculus courses.

Derivatives of Exponential Functions

Key things to remember

The slope of aˣ is a multiple of aˣ

The natural logarithm undoes eˣ

Worked examples

Where marks get lost

Flashcards 7

Practice