Exponential Functions (HSC SSCE Mathematics Advanced): Revision Notes

Investigation: Differentiation of Exponential Functions

This investigation explores one of the most remarkable properties in mathematics: Euler's number $e$ is the unique base for which an exponential function is its own derivative. Through numerical and graphical methods, you will discover why $\frac{d}{dx}(e^x) = e^x$ and understand what makes $e$ so special compared to other exponential functions.

Learning objectives

This investigation aims to help you develop understanding in three key areas. First, you will estimate the limit $\lim_{h \to 0} \frac{a^h - 1}{h}$ for various positive values of $a$ using technology, discovering that this limit equals 1 when $a = e$. Second, you will derive and verify that $\frac{d}{dx}(e^x) = e^x$, confirming that $e$ is unique in this property. Finally, you will use graphing software to visualise that $e^x$ is its own derivative and compare this behaviour with other exponential functions.

Exploring the limit

The special nature of Euler's number $e$ becomes apparent when we examine the derivative of exponential functions. For any exponential function $a^x$, the derivative involves a particular limit that determines how quickly the function grows. This limit is $\lim_{h \to 0} \frac{a^h - 1}{h}$, and it represents the gradient (slope) of the curve $y = a^x$ at the point where $x = 0$.

Understanding why this limit matters requires thinking about what differentiation means. When we differentiate, we measure the instantaneous rate of change. For exponential functions with different bases, this rate of change varies. The remarkable discovery is that when the base $a$ equals approximately 2.71828 (which is $e$), this limit equals exactly 1, creating a perfect simplification in the derivative formula.

Conducting the spreadsheet investigation

A systematic numerical approach reveals the pattern. Set up your spreadsheet to test different base values and observe how the limit behaves as $h$ becomes very small.

Step 1: Create a row with different base values. Enter values for $a$ including 2, 2.71828 (approximately $e$), and 3 in separate columns (for example, in cells B1, C1, and D1).

Step 2: In a column, list progressively smaller values of $h$. Use values such as:

• $h = 0.1$
• $h = 0.01$
• $h = 0.001$
• $h = 0.0001$

Step 3: Calculate the expression $\frac{a^h - 1}{h}$ for each combination of $a$ and $h$. For instance, if your base value is in cell B1 and your $h$ value is in cell A2, you would enter the formula $\frac{(\text{B1})^{\text{A2}} - 1}{\text{A2}}$ in cell B2. Copy this formula across and down to fill your table with values for all combinations.

Step 4: Observe and record the values in each column as $h$ approaches 0. Notice how the values stabilise (approach a limit) for each base $a$.

Step 5: Pay special attention to the column where $a \approx 2.71828$. Verify that the limit approaches 1. This observation is crucial because it suggests that this particular base value makes the derivative formula especially simple.

Investigation questions

Consider these questions to deepen your understanding:

1. What are the approximate limits for $a = 2$, $a = 3$, and $a = e$? Compare these values and note the pattern.

2. Why does $a = e$ yield a limit of 1? How does this relate to the derivative? Think about what happens when you multiply by 1 in the derivative formula.

Deriving the derivative

Now we use first principles to derive the derivative of exponential functions formally. This derivation shows exactly why the limit we investigated is so important.

The derivative of $f(x) = a^x$ using first principles begins with the definition:

$f'(x) = \lim_{h \to 0} \frac{a^{x+h} - a^x}{h}$

We can factorise the numerator by taking out $a^x$:

$f'(x) = \lim_{h \to 0} \frac{a^x \cdot a^h - a^x}{h}$

$= \lim_{h \to 0} \frac{a^x(a^h - 1)}{h}$

$= a^x \times \lim_{h \to 0} \frac{a^h - 1}{h}$

This final expression reveals the key insight: the derivative of $a^x$ equals $a^x$ multiplied by the limit we investigated earlier. For most bases, this limit is some constant value that complicates the derivative. However, when $a = e$, the limit equals 1, giving us the beautifully simple result:

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

Numerical verification

To verify this property numerically and compare it with other bases, follow these steps using a spreadsheet:

Step 1: For $x = 1$, compute $\frac{e^{1+h} - e^1}{h}$ for small values of $h$ (such as 0.1, 0.01, and 0.001). Compare these results to $e^1 \approx 2.71828$. You should find the values converge to approximately 2.71828, confirming that the derivative equals the function value.

Step 2: For comparison, calculate $\frac{2^{1+h} - 2^1}{h}$ for the same $h$ values. Compare your results to the limit you found in the previous section multiplied by $2^1$. Notice that the derivative of $2^x$ is not equal to $2^x$ itself but rather to $2^x$ times a constant factor.

Step 3: Use a graphing calculator to plot both $e^x$ and its derivative on the same axes. You should observe that the graphs coincide perfectly—they are the same curve. Then plot $2^x$ alongside $k \times 2^x$ (using your limit value for $k$) to see how they differ.

Investigation questions

Explore these questions to consolidate your understanding:

1. Does $\frac{e^{1+h} - e^1}{h}$ approach $e^1$ as $h$ approaches 0? Why is this significant? This confirms numerically what we proved algebraically.

2. How does the derivative of $2^x$ differ from $e^x$ graphically and numerically? Consider both the shape of the derivative curve and the actual numerical values.

3. Why do the graphs of $e^x$ and its derivative coincide? This is the visual representation of the self-derivative property.

Discussion

Reflecting on these investigations helps solidify the mathematical concepts and their implications:

1. Why is $\frac{d}{dx}(e^x) = e^x$ a unique property compared to other bases like $2^x$? Consider that for any base $a \neq e$, the derivative is $a^x$ times some constant factor. Only when $a = e$ does this constant equal 1, making the derivative exactly equal to the original function.

2. How did graphing $e^x$ and its derivative clarify its self-derivative property? The visual confirmation that the two curves are identical provides powerful intuitive understanding beyond the algebraic proof.

3. How does the limit $\lim_{h \to 0} \frac{e^h - 1}{h} = 1$ explain why $e^x$ is its own derivative? This limit is the "constant factor" in the derivative formula. When it equals 1, multiplying by it doesn't change the expression, so $\frac{d}{dx}(e^x) = e^x \times 1 = e^x$.

Remember!