Difficulty: Intermediate to Advanced. Prerequisites: Derivatives Fundamentals (Part 2 of these notes).
When functions are multiplied, divided, or nested inside each other, the basic rules are not enough. The product rule handles f(x)·g(x), the quotient rule handles f(x)/g(x), and the chain rule handles compositions like f(g(x)). The chain rule is the most important of the three and appears constantly. Logarithmic differentiation is a technique for complicated products, quotients, and powers where taking ln of both sides before differentiating saves substantial algebra.
Product rule
If y = F(x)·S(x), then y' = F'(x)·S(x) + F(x)·S'(x). Derivative of the first times the second, plus the first times the derivative of the second.
Think of it as: "Each factor gets a turn being differentiated while the other stays put."
Quotient rule
If y = T(x)/B(x), then y' = [T'(x)·B(x) − T(x)·B'(x)] / [B(x)]². Some students remember this as "lo d-hi minus hi d-lo, over lo squared."
Think of it as: "Bottom times derivative of top, minus top times derivative of bottom, all over bottom squared."
Chain rule
If y = E[I(x)] (a composite function with exterior function E and interior function I), then y' = E'[I(x)] · I'(x). Differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.
Think of it as: "Peel off the outer layer, then deal with the inner layer."
General power rule
A special case of the chain rule for y = [u(x)]^n. Then y' = n[u(x)]^(n−1) · u'(x).
In simple terms, this means: "Power rule as usual, but multiply by the derivative of whatever is inside the brackets."
Composite function
A function formed by plugging one function into another: f(g(x)). The inside function g(x) feeds its output into the outside function f.
Logarithmic differentiation
A technique where you take the natural log of both sides of y = f(x), use log properties to simplify, then differentiate implicitly. Especially useful when f(x) involves products, quotients, and powers of several terms.
Formula: if y = F(x)·S(x), then y' = F'(x)·S(x) + F(x)·S'(x).
Example: f(x) = 5x²(3x² − 5x + 2). Let F = 5x² and S = 3x² − 5x + 2. Then F' = 10x and S' = 6x − 5. So f'(x) = 10x(3x² − 5x + 2) + 5x²(6x − 5).
Example with exponentials: y = 4x³·e^(5x). Then y' = 12x²·e^(5x) + 4x³·(5e^(5x)) = e^(5x)(12x² + 20x³).
Example with ln: y = 6x⁵·ln x. Then y' = 30x⁴·ln x + 6x⁵·(1/x) = 30x⁴ ln x + 6x⁴.
Formula: if y = T(x)/B(x), then y' = [T'(x)·B(x) − T(x)·B'(x)] / [B(x)]².
Example: Q(x) = (x² − 5x + 7)/(2x). T = x² − 5x + 7, B = 2x, T' = 2x − 5, B' = 2. Then Q'(x) = [(2x−5)(2x) − (x²−5x+7)(2)] / (2x)².
Example: f(x) = e^(3x)/(x³ + 2). Then f'(x) = [3e^(3x)(x³+2) − e^(3x)(3x²)] / (x³+2)².
Example: tangent line to f(x) = (2x−5)/(2x−1) at x = 2. f(2) = −1/3. f'(x) = [2(2x−1) − (2x−5)(2)] / (2x−1)² = 8/(2x−1)². f'(2) = 8/9. Tangent: y + 1/3 = (8/9)(x − 2).
The chain rule is needed whenever one function is plugged into another.
Formula: if y = E[I(x)], then y' = E'[I(x)] · I'(x). Differentiate the outer function (keeping the inner function untouched), then multiply by the derivative of the inner function.
General power rule (chain rule for powers): if y = [u(x)]^n, then y' = n[u(x)]^(n−1) · u'(x).
Example: f(x) = (5x − 2)⁴. Inner: u = 5x − 2, u' = 5. Then f'(x) = 4(5x−2)³ · 5 = 20(5x−2)³.
Example: y = √(6x² + 1) = (6x²+1)^(1/2). Then y' = (1/2)(6x²+1)^(−1/2) · 12x = 6x/√(6x²+1).
Example: g(x) = 2/(x² − 3)⁴ = 2(x²−3)^(−4). Then g'(x) = 2(−4)(x²−3)^(−5) · 2x = −16x/(x²−3)⁵.
Chain rule with exponentials: d/dx[e^(u)] = e^(u) · u'. Example: y = e^(5−2x²). Then y' = e^(5−2x²) · (−4x) = −4x·e^(5−2x²).
Chain rule with logarithms: d/dx[ln(u)] = u'/u. Example: y = ln(x² − 7x + 1). Then y' = (2x − 7)/(x² − 7x + 1).
Combining rules: When a function involves both a product (or quotient) and a composition, apply the product/quotient rule at the outer level and the chain rule on the composed pieces.
Example: y = x²(2x − 5)³. Product rule first: y' = 2x(2x−5)³ + x² · 3(2x−5)² · 2 = 2x(2x−5)³ + 6x²(2x−5)².
Example: y = 5x² ln(3x − 1). Product rule: y' = 10x·ln(3x−1) + 5x² · 3/(3x−1).
Useful when y is a product, quotient, or power of several expressions, and applying the product/quotient rule directly would be unwieldy.
Steps:
Take ln of both sides: ln y = ln[f(x)].
Use log properties (product → sum, quotient → difference, power → coefficient) to expand.
Differentiate both sides. The left side becomes y'/y.
Multiply both sides by y (the original function) to isolate y'.
Example: y = x²·e^(−x)·(3x+5)³. Take ln: ln y = 2 ln x − x + 3 ln(3x+5). Differentiate: y'/y = 2/x − 1 + 9/(3x+5). Then y' = y · [2/x − 1 + 9/(3x+5)].
Algebraic properties of logarithms used throughout: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln(a^p) = p ln a.
\text{Product: } [F \cdot S]' = F'S + FS'\text{Quotient: } \left[\frac{T}{B}\right]' = \frac{T'B - TB'}{B^2}\text{Chain: } \frac{d}{dx}[E(I(x))] = E'(I(x)) \cdot I'(x)\text{General power: } \frac{d}{dx}[u^n] = n\,u^{n-1} \cdot u'\frac{d}{dx}[e^u] = e^u \cdot u' \qquad \frac{d}{dx}[\ln u] = \frac{u'}{u}Students often try to differentiate a product by simply differentiating each factor separately: d/dx[f·g] ≠ f'·g'. You must use the product rule.
Students often mix up the sign in the quotient rule. The numerator is T'B minus TB' (not the other way round). Getting the subtraction order wrong flips the sign of the answer.
Students often forget the chain rule's "multiply by the derivative of the inside" step. This is the single most common error in differentiation. If the inside is anything other than plain x, you need the chain rule.
Students often apply the chain rule to the wrong layer. Identify the outermost operation first (the exterior function) and the innermost expression (the interior function) before differentiating.
⚠️ The chain rule appears in almost every differentiation problem from this point onwards. Practise identifying the inside and outside functions until it is automatic.
⚠️ Finding where the tangent line is horizontal means setting f'(x) = 0. When f'(x) involves a product (e.g., after the product rule), set each factor to zero separately. Remember that e^(anything) is never zero.
⚠️ Quotient rule applications: expect word problems involving functions like N(t) = 180t/(t+4), where you find N'(t) and interpret N(16) and N'(16) in context.
⚠️ Logarithmic differentiation: likely to appear as a "use logarithmic differentiation to find f'(x)" directive. Know the three log properties cold.
Fill in the blank: d/dx[f(x)·g(x)] = f'(x)·g(x) + __________.
True or false: d/dx[ln(x² + 1)] = 1/(x² + 1).
Fill in the blank: For y = (3x + 7)⁵, y' = 5(3x + 7)⁴ · __________.
True or false: e^(2x) can equal zero for some value of x.
Answers: 1. f(x)·g'(x). 2. False (it is 2x/(x²+1), because of the chain rule). 3. 3 (the derivative of the inside). 4. False (e raised to any power is always positive).
Q: Differentiate y = 4x³·e^(5x).
A: Product rule. y' = 12x²·e^(5x) + 4x³·(5e^(5x)) = e^(5x)(12x² + 20x³) = 4x²·e^(5x)(3 + 5x).
Q: Differentiate f(x) = e^(3x)/(x³ + 2).
A: Quotient rule. f'(x) = [3e^(3x)(x³+2) − e^(3x)(3x²)] / (x³+2)² = e^(3x)[3(x³+2) − 3x²] / (x³+2)² = e^(3x)(3x³ − 3x² + 6) / (x³+2)².
Q: Find the value(s) of x where the tangent line is horizontal on f(x) = (x − 3)e^(2x).
A: f'(x) = (1)e^(2x) + (x−3)(2e^(2x)) = e^(2x)[1 + 2x − 6] = e^(2x)(2x − 5). Set f'(x) = 0: e^(2x) ≠ 0, so 2x − 5 = 0, x = 5/2.
Q: Differentiate y = ln(x² − 7x + 1).
A: Chain rule. y' = (2x − 7)/(x² − 7x + 1).
Q: Use logarithmic differentiation to find y' for y = x³(2x−1)³ / (4x² − 3).
A: ln y = 3 ln x + 3 ln(2x−1) − ln(4x²−3). Differentiate: y'/y = 3/x + 6/(2x−1) − 8x/(4x²−3). Then y' = y·[3/x + 6/(2x−1) − 8x/(4x²−3)].
These rules are essential for graph analysis (Part 4), where you set the first derivative to zero to find critical points and use the second derivative for concavity. The chain rule also reappears in integration (substitution method) later in the course. Marginal analysis problems from Part 2 become more complex once the revenue or cost function involves products or quotients.
Tags: product rule, quotient rule, chain rule, general power rule, composite function, inside function, outside function, logarithmic differentiation, log properties, horizontal tangent line, derivative of e^u, derivative of ln u, MAC 2233, business calculus, FIU