Exponential and Log Derivatives: Step-by-Step Guide
The natural exponential e^x is its own derivative, and the natural log has derivative 1/x. For general bases, d/dx[a^x] = a^x ln(a) and d/dx[log_a(x)] = 1/(x ln(a)). Combine these with the chain rule whenever the exponent or argument is a function of x.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Recognize the exponential or log form. Identify whether the function is e^(g(x)), a^(g(x)), ln(g(x)), or log_a(g(x)). Note both the base and the inner expression g(x).\[y = e^{g(x)}, \; a^{g(x)}, \; \ln(g(x)), \; \log_a(g(x))\]
- Apply the base derivative formula. Use d/dx[e^u] = e^u, d/dx[a^u] = a^u ln(a), d/dx[ln u] = 1/u, d/dx[log_a u] = 1/(u ln(a)).\[\frac{d}{dx}[e^u] = e^u, \;\; \frac{d}{dx}[\ln u] = \frac{1}{u}\]
- Multiply by the inner derivative (chain rule). Multiply the result by g'(x), the derivative of the inner expression. This is the chain rule applied to exponentials and logs.\[\frac{d}{dx}[e^{g(x)}] = e^{g(x)} \cdot g'(x)\]
- Simplify the final expression. Combine constants, factor common terms, and write the answer as cleanly as possible.\[\frac{dy}{dx} = \text{simplified}\]
Worked example
Find d/dx of ln(x^2 + 1) + e^(3x).
Step 1: The function has two terms. Term A: ln(x^2 + 1) with inner g(x) = x^2 + 1. Term B: e^(3x) with inner h(x) = 3x.
Step 2: For ln, the rule is d/dx[ln u] = u'/u. So d/dx[ln(x^2 + 1)] = (2x)/(x^2 + 1).
Step 3: For e^u, the rule is d/dx[e^u] = e^u * u'. So d/dx[e^(3x)] = e^(3x) * 3 = 3e^(3x).
Step 4: Add the term derivatives: dy/dx = (2x)/(x^2 + 1) + 3e^(3x).
Step 5: This is fully simplified.
Verification: At x = 0, dy/dx = 0/(1) + 3e^0 = 0 + 3 = 3. Numerical check: (ln(0.001^2+1) + e^(0.003) - ln(1) - e^0)/0.001 approximately (0.000001 + 0.0030045)/0.001 approximately 3.0055, close to 3.
Tips and common mistakes
- Memorize the four base formulas; everything else is the chain rule applied to them.
- The natural choice for calculus is base e, since d/dx[e^x] = e^x and d/dx[ln x] = 1/x have no extra constants.
- For a^x with a not equal to e, the constant ln(a) appears: d/dx[a^x] = a^x ln(a).
- For ln(g(x)), the derivative is g'(x)/g(x) — no chain-rule multiplier is forgotten this way.
- Use log properties (ln(ab) = ln a + ln b, ln(a^n) = n ln a) to simplify before differentiating; this often dodges the quotient rule.
- Treat e and pi as constants; their derivatives are 0 unless they appear as exponents on a variable.
- For functions like x^x, neither power rule nor exponential rule applies — use logarithmic differentiation.
- Always check the inner function is in the domain (positive for ln) at the point where you evaluate.