Derivatives of Exponential and Logarithmic Functions

The functions e^x and ln x are the two most important functions in calculus, and their derivatives have a beautiful simplicity: the derivative of e^x is e^x itself. This is the only function (apart from the zero function) that is its own derivative, and it is the reason exponential functions model so many natural processes where the rate of change is proportional to the current amount.

Knowing how to differentiate exponentials and logarithms matters because they appear in compound interest, population growth, radioactive decay, pH calculations, and the chain rule problems of almost every calculus exam. The few rules here unlock all of them, and each rule has a logical reason behind it rather than being a formula to memorize blindly.

Key Concepts

• Derivative of e^x: d/dx(e^x) = e^x. The number e (about 2.71828) is defined precisely so that this rule holds.
• Derivative of a^x: d/dx(a^x) = a^x · ln a, where a is a positive constant. The extra factor ln a adjusts for the different base.
• Derivative of ln x: d/dx(ln x) = 1/x for x greater than 0. This is the natural logarithm, base e.
• Derivative of log base a: d/dx(log_a x) = 1 / (x · ln a), which follows from the change-of-base formula log_a x = ln x / ln a.
• Chain rule versions: d/dx(e^u) = e^u · du/dx and d/dx(ln u) = (1/u) · du/dx, where u is a function of x.

Formulas

• d/dx(e^x) = e^x — no extra factors; SI units of the result match the context of the problem.
• d/dx(a^x) = a^x ln a — assumes a is a positive constant, a not equal to 1.
• d/dx(ln x) = 1/x — valid for x greater than 0. For negative x, d/dx(ln |x|) = 1/x also holds.
• d/dx(log_a x) = 1/(x ln a) — assumes a is a positive constant, a not equal to 1, x greater than 0.

Worked Example

Find the derivative of y = x² · e^x and evaluate the slope at x = 1.

1. Step 1: Use the product rule. Let u = x² and v = e^x, so y' = u'v + uv'.
2. Step 2: Differentiate each part: u' = 2x and v' = e^x.
3. Step 3: Substitute: y' = 2x · e^x + x² · e^x.
4. Step 4: Factor out e^x: y' = e^x (x² + 2x).
5. Step 5: Evaluate at x = 1: y' = e¹ (1 + 2) = 3e, which is about 3 × 2.71828 = 8.15484.

Common Misconceptions

• "The derivative of ln x is 1 / ln x." Wrong — it is 1/x. The logarithm itself disappears completely after differentiation.
• "The derivative of a^x is x · a^(x−1)." That is the power rule, which only works when the exponent is a constant. When the variable is in the exponent, the rule is a^x ln a.
• "The derivative of x^x is x · x^(x−1)." Neither rule applies directly since both base and exponent vary. This needs logarithmic differentiation, giving x^x (ln x + 1).

Real-World Applications

• Population biology: if a bacterial colony grows as P = P₀ e^(kt), the growth rate is dP/dt = kP, so the derivative tells you how fast the colony is expanding right now.
• Radioactive decay: the decay rate of a sample is the derivative of its mass function m = m₀ e^(−λt), giving −λm.
• Finance: continuous compounding uses A = Pe^(rt), and dA/dt shows how quickly the account balance is growing at any instant.
• Chemistry: the pH scale is logarithmic, and small changes in hydrogen ion concentration are analyzed with the derivative of the logarithm.

To recap: e^x differentiates to itself, a^x picks up a factor of ln a, and ln x differentiates to 1/x. Combined with the product rule and chain rule, these four rules handle every exponential and logarithmic derivative you will meet.