Exponential & Logarithmic Derivatives

Exponential and logarithmic functions appear throughout mathematics, science, and engineering — from modelling population growth and radioactive decay to calculating compound interest and signal processing. Knowing how to differentiate these functions is an essential skill in calculus, allowing us to analyse rates of change in any situation where quantities grow or shrink multiplicatively. The natural exponential function f(x) = eˣ has the remarkable property that it is its own derivative, making it uniquely important in differential equations and mathematical modelling. Logarithmic differentiation — using the natural logarithm to simplify complex products, quotients, and variable-exponent expressions before differentiating — is a powerful technique that extends far beyond simply differentiating log functions themselves. In this article you will derive and apply the key differentiation rules for eˣ, aˣ, ln x, and logₐ x, understand why the number e is so special, and practise logarithmic differentiation as a general strategy. Mastering these rules connects directly to integration, differential equations, and real-world modelling across every quantitative discipline.

Key takeaways
  • Why e is special — the limit definition
  • Derivative of the general exponential aˣ
  • Derivative of the natural logarithm
  • Derivative of logₐ x
  • Logarithmic differentiation
  • Chain rule combinations

Why e is special — the limit definition

The number e ≈ 2.71828 is defined so that the slope of y = eˣ at every point equals the function value itself. Formally, e = lim(n→∞)(1 + 1/n)ⁿ. From the limit definition of the derivative, d/dx[eˣ] = lim(h→0) (e^(x+h) − eˣ)/h = eˣ · lim(h→0) (eʰ − 1)/h. The crucial limit lim(h→0)(eʰ − 1)/h = 1 (proved from the definition of e), giving d/dx[eˣ] = eˣ. No other base has this clean property without an extra constant factor.

Derivative of the general exponential aˣ

For any positive base a ≠ 1, write aˣ = e^(x ln a) using the identity a = e^(ln a). Applying the chain rule: d/dx[aˣ] = d/dx[e^(x ln a)] = e^(x ln a) · ln a = aˣ ln a. Notice that when a = e, ln e = 1 and we recover d/dx[eˣ] = eˣ. The factor ln a measures how far the base a differs from e.

Derivative of the natural logarithm

Let y = ln x, so eʸ = x. Differentiating both sides implicitly with respect to x: eʸ · dy/dx = 1, so dy/dx = 1/eʸ = 1/x. Therefore d/dx[ln x] = 1/x for x > 0. Combined with the chain rule: d/dx[ln(u)] = u'/u, which is extremely useful for simplifying complicated expressions.

Derivative of logₐ x

Using the change-of-base formula, logₐ x = ln x / ln a. Since ln a is a constant, d/dx[logₐ x] = (1/ln a) · d/dx[ln x] = 1/(x ln a). Again, setting a = e gives 1/(x · 1) = 1/x, confirming the natural log result.

Logarithmic differentiation

When differentiating y = [f(x)]^[g(x)] — a variable raised to a variable power — take the natural log of both sides first: ln y = g(x) · ln[f(x)]. Differentiate implicitly: (1/y)(dy/dx) = g'(x) ln[f(x)] + g(x) · f'(x)/f(x). Multiply both sides by y to get dy/dx. This technique also simplifies products and quotients with many factors: take ln of both sides, differentiate, then multiply by y. For example, y = (x²)(sin x)(eˣ) becomes ln y = 2 ln x + ln(sin x) + x, giving (1/y)(dy/dx) = 2/x + cos x/sin x + 1.

Chain rule combinations

Most exam problems combine these rules with the chain rule. Key patterns: d/dx[e^(f(x))] = f'(x)·e^(f(x)); d/dx[ln(f(x))] = f'(x)/f(x); d/dx[a^(f(x))] = f'(x)·a^(f(x))·ln a. Always identify the outer function and inner function before differentiating. For example, d/dx[e^(3x²)] = 6x·e^(3x²) because the inner function is 3x² with derivative 6x.

The Key Equation

ddx[ax]=axln⁡a,ddx[log⁡ax]=1xln⁡a\frac{d}{dx}[a^x] = a^x \ln a, \qquad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}
  • a — Base (dimensionless): Any positive real number not equal to 1; the base of the exponential or logarithm.
  • \ln a — Natural logarithm of a (dimensionless): The logarithm of a to base e; acts as a scaling factor measuring how far the base a is from e. When a = e, ln a = 1 and both formulas reduce to the standard eˣ and ln x rules.
  • x — Independent variable (dimensionless): The input variable with respect to which we differentiate.

Worked Example

Problem. Differentiate y = x^x using logarithmic differentiation.

dydx=xx(ln⁡x+1)\frac{dy}{dx} = x^x(\ln x + 1)

Answer. dy/dx = x^x (ln x + 1)

Figure
Diagram
y = eˣ and y = ln x are reflections across y = xxyy = eˣy = ln xy = x(0,1)(1,0)Slope of eˣ at any point equals its y-value; slope of ln x at x equals 1/x
Origins · 1 minLeonhard Euler and how this was discovered
LE
Leonhard Euler
1707–1783 · Swiss

Euler introduced the notation e for the base of the natural exponential and established its fundamental properties, including the identity eˣ = lim(n→∞)(1 + x/n)ⁿ and the famous Euler's formula e^(iπ) + 1 = 0. His systematic treatment of exponential and logarithmic functions in 'Introductio in Analysin Infinitorum' (1748) laid the groundwork for all modern analysis of these functions. Euler lost sight in his right eye in 1738 and became almost totally blind by 1771, yet he continued to produce roughly half of his total mathematical output after losing his sight, reportedly saying his productivity increased because he had fewer distractions.

  • The Chain Rule — Nearly every exponential and logarithmic derivative in practice requires the chain rule when the exponent or argument is itself a function of x.
  • Implicit Differentiation — The derivative of ln x is derived via implicit differentiation of eʸ = x, and logarithmic differentiation relies on differentiating ln y implicitly.
  • Limit Definition of the Derivative — Understanding why d/dx[eˣ] = eˣ requires evaluating lim(h→0)(eʰ−1)/h = 1 from the limit definition.
  • Related Rates — Exponential growth and decay models (e.g. population, radioactive decay) appear in related-rates problems and require these derivative rules.