Source: Warmup 3.2, Calculus for Business (FIU)
Tags: exponential derivatives, logarithmic derivatives, ln x derivative, e^x derivative, log properties, calculus for business, FIU, Section 3.2
The derivative of ln(x) is 1/x and the derivative of e^x is e^x. Logarithm properties let you break apart products, quotients and powers inside a log, which simplifies differentiation. Most problems in this section combine these two rules with the power rule you already know.
Difficulty: Intermediate
Prerequisites: Power rule for derivatives (Section 3.1), basic properties of exponents and logarithms from algebra.
Up to this point in Calculus for Business you have been differentiating polynomials with the power rule. Section 3.2 introduces two new function families, exponential and logarithmic, that show up constantly in business models: compound interest, continuous growth, depreciation, demand curves. You need to be comfortable with basic exponent and log algebra from your prerequisites, because the derivative rules themselves are short but the simplification steps lean heavily on log properties.
Natural logarithm, ln(x)
The logarithm with base e. Written ln(x) rather than log_e(x). Defined only for x > 0. Think of it as: "the power you raise e to in order to get x."
Euler's number, e
The irrational constant approximately equal to 2.71828. It is the base of the natural logarithm and the natural exponential function. In simple terms, e is the number that makes continuous-growth maths work cleanly.
Exponential function, e^x
The function f(x) = e^x. It is its own derivative, which makes it unique among all functions. Think of it as: the function whose rate of change at any point equals its own value at that point.
Logarithm base b, log_b(x)
The power to which b must be raised to produce x. So if log_b(x) = m, then b^m = x. In simple terms, a logarithm answers the question "b to what power gives me x?"
Common logarithm, log(x)
The logarithm with base 10, sometimes written log_10(x). In calculus for business it appears less often than ln(x), but the conversion formula lets you move between bases.
If f(x) = ln(x), then f'(x) = 1/x
The derivative shrinks as x grows: at x = 1 the slope is 1, at x = 10 the slope is 0.1
This rule applies only when the argument is plain x. When the inside is more complex (e.g. ln(x^4)), use log properties to simplify first, then differentiate
If f(x) = e^x, then f'(x) = e^x
e^x is the only function that is its own derivative. This is why e is special
A constant multiplier carries through: d/dx [2e^x] = 2e^x
Product rule for logs: log_b(u · v) = log_b(u) + log_b(v)
A log of a product splits into a sum of logs
Power rule for logs: log_b(x^r) = r · log_b(x)
The exponent pulls down in front as a multiplier. This is the property you will use most often in this section
Quotient rule for logs: log_b(u / v) = log_b(u) - log_b(v)
A log of a fraction splits into a difference of logs
e^(ln x) = x (the exponential and natural log undo each other)
ln(e^x) = x (same idea, other direction)
If a^x = m, take ln of both sides: x · ln(a) = ln(m), so x = ln(m) / ln(a)
If log_b(x) = m, rewrite in exponential form: b^m = x
Example (a): f(x) = 2e^x + 3 ln(x^7) - 1112/29
Simplify first: 3 ln(x^7) = 3 · 7 · ln(x) = 21 ln(x) using the power rule for logs
Differentiate term by term: f'(x) = 2e^x + 21/x + 0
The constant -1112/29 vanishes
Example (b): f(x) = x^2 - ln(x) + 2e^x
Differentiate term by term: f'(x) = 2x - 1/x + 2e^x
Each term uses a different rule: power rule, ln rule, exponential rule
Example (c): f(x) = ln(x^4) + e^x - 45
Simplify first: ln(x^4) = 4 ln(x) using the power rule for logs
Differentiate: f'(x) = 4/x + e^x + 0
\frac{d}{dx}[\ln(x)] = \frac{1}{x}\frac{d}{dx}[e^x] = e^x\log_b(u \cdot v) = \log_b(u) + \log_b(v)\log_b(x^r) = r \cdot \log_b(x)\log_b\!\left(\frac{u}{v}\right) = \log_b(u) - \log_b(v)e^{\ln x} = x \quad \text{and} \quad \ln(e^x) = xa^x = m \implies x = \frac{\ln(m)}{\ln(a)}Students often think the derivative of ln(x) is 1/x times the derivative of x, and write (1/x) · 1 every time. That works when the argument is plain x, but when it is something like ln(x^4), you need to simplify with log properties first (or later, use the chain rule). Skipping this step is the most common error on exams
Students sometimes believe that d/dx [e^x] = x · e^(x-1), applying the power rule to the exponent. The power rule applies to x^n, not to e^x. These are different families of functions
Confusing ln and log. In most calculus textbooks, ln means base e and log without a subscript means base 10. Your professor may use log to mean ln in some contexts, so check the convention for your course
Forgetting that constants disappear. When a function ends in a constant term (like -45 or -1112/29), its derivative is zero. Students sometimes carry the constant through or try to differentiate it
Exponential functions model anything that grows or decays at a rate proportional to its current size: compound interest, population growth, radioactive decay, drug metabolism. The derivative tells you the instantaneous rate of change, which in a business context might be how fast revenue is growing at a specific moment.
Logarithmic functions appear in economics as utility functions (diminishing returns) and in finance when solving for time-to-double or interest rates. Differentiating a log-transformed model is a standard step in econometrics and data analysis.
⚠️ You will almost certainly be asked to differentiate a function that mixes polynomial, exponential and logarithmic terms. Be ready to apply all three rules in one problem
⚠️ Simplify ln(x^n) to n · ln(x) before differentiating. Examiners test whether you know to do this
⚠️ Know the log properties cold. Fill-in-the-blank questions on these identities are easy marks if you have memorised them, and costly if you have not
⚠️ The constant term trap: always check whether a term is a constant. d/dx[-1112/29] = 0, not -1112/29
True or False: The derivative of e^x is x · e^(x-1). (False. d/dx [e^x] = e^x.)
Fill in the blank: d/dx [ln(x)] = ______. (1/x)
True or False: ln(x^5) can be rewritten as 5 · ln(x). (True, by the power rule for logarithms.)
Fill in the blank: log_b(u · v) = ______. (log_b(u) + log_b(v))
True or False: The derivative of a constant, such as -45, is -45. (False. The derivative of any constant is 0.)
Q: Find the derivative of f(x) = 5e^x + ln(x^3) - 7.
A: Simplify ln(x^3) to 3 ln(x). Then f'(x) = 5e^x + 3/x.
Q: If log_b(x) = m, what is x in terms of b and m?
A: x = b^m. You rewrite from logarithmic form to exponential form.
Q: Find the derivative of f(x) = x^5 - 4 ln(x) + e^x + 100.
A: f'(x) = 5x^4 - 4/x + e^x. The constant 100 drops out.
Q: Rewrite ln(x^6) using logarithm properties, then differentiate.
A: ln(x^6) = 6 ln(x). The derivative is 6 · (1/x) = 6/x.
Q: Why is d/dx [e^x] different from d/dx [x^n]?
A: In x^n the base is the variable and the exponent is constant, so the power rule applies (n · x^(n-1)). In e^x the base is the constant e and the exponent is the variable, which gives a completely different rule: d/dx [e^x] = e^x.
This material connects directly to the chain rule (Section 3.3 or 3.4 in most business calculus courses), where you will differentiate compositions like ln(3x + 1) or e^(2x). The rules from Section 3.2 become the "outer" derivative in every chain rule problem involving logs or exponentials.
Exponential and logarithmic derivatives also appear in optimisation problems later in the course, particularly when modelling revenue, cost or profit functions that involve continuous growth or decay.
In statistics and econometrics courses, log-transformations of data are standard, and understanding d/dx [ln(x)] = 1/x is essential for interpreting coefficients in log-linear regression models.
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