Wemazi

Topic 2.7

Differentiate exponential functions

Builds on differentiating sine and cosine. 6 minutes.

How steep is an exponential curve?

The curve is \(y = 2^x\). It is green because it rises, slowly on the left and quickly on the right.

The orange line is the tangent, and its slope is the number to find. The orange bar on the top graph shows the height of the curve, and the orange bar on the lower graph shows the slope.

Go to \(x = 0\). The height is 1 and the slope is about 0.69. Go to \(x = 2\). The height is 4 and the slope is about 2.77.

The last number below the figure is the slope divided by the height. It stays at about 0.69 as you move the point, so the slope is always 0.69 times the height.

Changing the base

Move the base slider to change the curve \(y = a^x\), where the base \(a\) is a positive number.

For each base, the slope is a fixed multiple of the height. The multiple depends on \(a\) and not on \(x\), so the last number below the figure stays the same while the point moves.

At the base 1, the curve is a flat line, so the multiple is 0. As you raise the base, the multiple grows. At the base \(e\), which is about 2.72, the multiple is 1, so the slope equals the height at every point.

If the base is below 1, the curve falls and is red, so the multiple and the slope are negative.

Notation

  1. For a base \(a > 0\), the slope of \(a^x\) is the height \(a^x\) times the multiple \(\ln a\).

    \[\frac{d}{dx}\,a^x = \cy{a^x \ln a}\]

    The symbol \(\ln a\) means the natural logarithm of \(a\). It is the power to which \(e\) must be raised to give \(a\).

  2. For the base \(e\), the multiple is \(\ln e = 1\), so the slope of \(e^x\) equals its height.

    \[\frac{d}{dx}\,e^x = \cy{e^x}\]

  3. As a check, \(\ln 2 \approx 0.69\), which is the multiple in the opening figure. For the base 3, \(\ln 3 \approx 1.10\), so the slope is 1.1 times the height.

Example 1: a base above 1

Find the slope of \(y = 3^x\) at \(x = 2\).

  1. The base is 3, so the slope is the height \(3^x\) times \(\ln 3\).

    \[y' = 3^x \ln 3\]

  2. Substitute \(x = 2\). The height is \(3^2 = 9\).

    \[y'(2) = 3^2\ln 3 = \cy{9\ln 3}\]

  3. As a check, \(9\ln 3 \approx 9.89\), which is the height of the orange bar on the lower graph.

Example 2: a base below 1

Find the slope of \(y = \left(\frac12\right)^x\) at \(x = 1\).

  1. The base is below 1, so the curve is red and falls. The slope must be negative.

  2. A tempting wrong move: Use the power rule: bring the exponent down and subtract 1 from it.

    \[\cr{y' = x\left(\tfrac12\right)^{x-1}}, \qquad \cr{y'(1) = 1}\]

    This gives a positive slope, as the red dashed line shows, but the curve falls at \(x = 1\). The power rule is for \(x^n\), where the exponent is a fixed number. Here the exponent is the variable.

  3. Use the rule for \(a^x\) with \(a = \frac12\).

    \[y' = \left(\tfrac12\right)^x \ln\tfrac12\]

  4. At \(x = 1\), the height is \(\frac12\), and \(\ln\frac12 = -\ln 2\) because \(e^{-\ln 2} = \frac12\).

    \[y'(1) = \tfrac12\ln\tfrac12 = \cy{-\tfrac12\ln 2}\]

  5. As a check, the slope is about \(-0.35\), so the orange bar on the lower graph lies below the axis.

Practice

Find the slope of \(y = 5^x\) at \(x = -1\).

Move the point to \(x = -1\). The curve is low and rises slowly there, so the slope is a small positive number.

Solution

  1. The slope is the height \(5^x\) times \(\ln 5\).

    \[y' = 5^x\ln 5\]

  2. Substitute \(x = -1\). The height is \(5^{-1} = \frac15\).

    \[y'(-1) = 5^{-1}\ln 5 = \cy{\frac{\ln 5}{5}}\]

  3. As a check, the slope is about 0.32. It is small and positive, as the picture shows.

Questions

Questions 1 and 2 use an exponential curve with its slope graph below. Question 3 needs a calculation.

  1. The marked point is on \(y = e^x\), where the height is \(e \approx 2.72\). What is the slope of the tangent?

    • \(e\)
    • 1
    • 0
    Answer

    \(e\)

  2. The marked point is on \(y = 2^x\) at \(x = 3\), where the height is 8. Which number is closest to the slope of the tangent?

    • 5.5
    • 8
    • 12
    • 0.69
    Answer

    5.5

  3. What is the derivative of \(10^x\)?

    • \(10^x\ln 10\)
    • \(10^x\)
    • \(x\cdot 10^{x-1}\)
    • \(\dfrac{10^x}{\ln 10}\)
    Answer

    \(10^x\ln 10\)

Summary

You differentiate \(a^x\) by multiplying it by \(\ln a\), and because \(\ln e = 1\), \(e^x\) is its own derivative.

The rule works because the slope of an exponential curve is a fixed multiple of its height, and \(\ln a\) is that multiple.

This rule is for a variable exponent, and the power rule is for a variable base with a fixed exponent.

This is one of four skills in topic 2.7 of Which & Why: AP Calculus.