Instant step-by-step derivatives using (f·g)′ = f′g + fg′ — enter any product, get every step from factor identification to final simplification
📅 Updated March 19, 2026⏱️ 5 min read🎓 AP Calculus · University Calculus I–III✅ Dual Engine · 13 Variables
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What Is the Product Rule? A Complete Calculus Guide
The product rule is one of the most fundamental differentiation formulas in calculus. It answers the question: how do you find the derivative of a function that is the product of two other functions? Whether you are differentiating x²·sin(x), eˣ·cos(x), or x·ln(x), the product rule is the key technique.
Formally, if h(x) = f(x)·g(x), the product rule formula states:
The Product Rule Formula — Leibniz Notation
h'(x) = f'(x)·g(x) + f(x)·g'(x)
In plain language: differentiate the first function and multiply by the second unchanged, then add the first function unchanged multiplied by the derivative of the second. A useful memory device is the phrase "derivative of first times second plus first times derivative of second" — or the UV rule used in British and South Asian curricula: (UV)' = U'V + UV'.
The product rule was first published by Gottfried Wilhelm Leibniz in 1684, who noticed that the differential of a product d(fg) is not simply df·dg but rather g·df + f·dg. Isaac Newton arrived at the same result independently. This rule laid a cornerstone of differential calculus applied daily in physics, engineering, and data science.
Why You Cannot Simply Multiply the Derivatives
The single most common misconception is writing (fg)' = f'·g'. This is always wrong. Counterexample: let h(x) = x·x = x². The correct derivative is 2x. Multiplying individual derivatives gives 1·1 = 1 — completely wrong. The product rule correctly gives 1·x + x·1 = 2x.
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Critical Error to Avoid: (f·g)' ≠ f'·g'. The correct formula always produces two terms: (f·g)' = f'·g + f·g'. Use this product rule derivative calculator above to verify your hand calculations instantly.
The Generalised Product Rule for Three or More Functions
For h(x) = f(x)·g(x)·k(x):
Generalised Product Rule — Three Functions
h'(x) = f'·g·k + f·g'·k + f·g·k'
Differentiate one factor at a time while leaving all others unchanged, then sum all resulting terms. Our product rule solver automatically detects two-factor, three-factor, or more products via abstract syntax tree analysis. For partial derivatives of product functions, use our dedicated partial derivative calculator.
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Professor's Insight: Before calculating, count your factors. That count tells you exactly how many terms your final derivative will contain — a quick self-check that catches most errors before they happen.
Product Rule Examples — Five Worked Problems
Mastering the product rule derivative requires practising with varied function types. These five examples progress from basic to triple-factor expressions requiring the product rule and chain rule simultaneously. All results can be verified with the derivative product rule calculator above.
Practice Strategy: Work each example by hand first, then enter it into the calculator and compare your steps. Discrepancies almost always reveal a sign error, a forgotten chain rule application, or an incomplete simplification.
Students frequently apply the wrong rule because they misidentify the structure of an expression. The distinction is architectural: are the functions multiplied side by side, or is one function nested inside another?
📦 Product Rule — Side by Side
Use when: h(x) = f(x) · g(x) Functions sit next to each other as separate factors.
Formula: h'(x) = f'g + fg'
🔗 Chain Rule — Nested Inside
Use when: h(x) = f(g(x)) One function is the argument of another.
Formula: h'(x) = f'(g(x)) · g'(x)
Expression
Rule(s) Required
Reason
x² · sin(x)
Product Rule
Two separate factors multiplied
sin(x²)
Chain Rule
x² is nested inside sin()
eˣ · cos(x)
Product Rule
Two separate factors multiplied
e^(cos x)
Chain Rule
cos(x) is nested inside the exponent
x² · sin(x²)
Both
Product rule for ×; chain rule for sin(x²)
√x · ln(x)
Product Rule
Two separate factors (sqrt is a power)
x · e^(x²)
Both
Product rule for ×; chain rule for e^(x²)
For dedicated chain rule calculations, visit the Chain Rule Calculator. Our product rule differentiation calculator handles combined product-and-chain expressions automatically.
5 Common Product Rule Mistakes — and How to Fix Them
Mistake 1 — Multiplying the Derivatives
Writing (fg)' = f'·g' is by far the most frequent error. Always write out both terms: f'g + fg'.
Mistake 2 — Omitting the Chain Rule on Composite Factors
When a factor is a composite function — sin(x²), e^(3x), (x+1)⁴ — apply the chain rule when differentiating it.
Mistake 3 — Applying the Product Rule When Not Needed
If one factor is a constant: d/dx[3·f(x)] = 3·f'(x). No product rule needed. Treat constants as constants.
Mistake 4 — Stopping Before Simplifying
After applying the product rule, always check for a common factor. For example, 2xeˣ + x²eˣ should be written as xeˣ(x+2).
Mistake 5 — Sign Errors with Trigonometric Derivatives
Remember: the derivative of cos(x) is −sin(x) (negative). This is the most common source of wrong final answers in product rule problems involving trigonometric functions.
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Exam Strategy: After computing your derivative, substitute x = 1 into both your hand answer and the calculator's result. Matching numerical values confirm correctness. Use this product rule to find derivative calculator for all exam preparation.
How to Use This Calculator & Real-World Applications
Using the Product Rule Calculator
Enter any product expression using standard notation: * for multiplication, ^ for exponents, log(x) for natural logarithm, and sqrt(x) for square root. Select your differentiation variable from the 13 available options, then press Enter or click the button.
Two engine modes are available. 🚀 Fast Mode delivers instant results using a high-performance symbolic computation engine. 🎯 Accurate Mode uses a professional-grade symbolic mathematics system used in academic research and computational science, for maximum precision and deeper simplification. Accurate Mode loads on demand and runs entirely in your browser — first use takes approximately 10–20 seconds.
Five export options appear after calculation: Print / PDF, Save as Text, Copy Result (LaTeX notation), Share URL (direct link with your expression), and New Calculation.
Real-World Applications of the Product Rule
Physics — Variable Mass Systems: Momentum p(t) = m(t)·v(t). When mass varies with time (rocket expelling propellant), p'(t) = m'(t)·v(t) + m(t)·v'(t) — the variable-mass equation of motion.
Economics — Marginal Revenue: Revenue R(x) = p(x)·q(x). Marginal revenue requires differentiating this product — central to every profit-maximisation problem in microeconomics.
Biology — Epidemic Models: In SIR models, the infection rate is β·S(t)·I(t). Analysing rates of change requires differentiating these product terms.
Engineering — Signal Processing: Amplitude-modulated signals: s(t) = A(t)·cos(ωt). Differentiating this product is required for instantaneous frequency analysis.
The "add and subtract a bridge term" technique also appears in the proofs of the quotient rule and chain rule. See our Quotient Rule Calculator for the analogous derivation.
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Professor's Note: Memorise the "add and subtract f(x)·g(x+Δx)" step — it is the single non-obvious move the entire proof depends on. Everything else follows mechanically from the definition.
Product Rule with Specific Function Types
Polynomial × Trigonometric Functions
For h(x) = xⁿ·sin(x): result is n·xⁿ⁻¹·sin(x) + xⁿ·cos(x). The polynomial factor decreases in degree, while the trig factor cycles. These patterns are used extensively in integration by parts.
Exponential × Trigonometric Functions
Products like eˣ·sin(x) and eˣ·cos(x) appear in damped oscillation models. Key results: (eˣ·sin x)' = eˣ(sin x + cos x) and (eˣ·cos x)' = eˣ(cos x − sin x).
Polynomial × Logarithmic Functions
d/dx[ln x] = 1/x. For xⁿ·ln x: result is xⁿ⁻¹(n·ln x + 1). For the simple case x·ln x (entropy calculations): derivative = ln x + 1. Enter x * log(x) in the log product rule calculator above to verify.
Radical × Logarithmic Functions
For √x·ln x = x^(1/2)·ln x: result is (ln x + 2)/(2√x). This simplification — pulling out a common radical denominator — is a mark of strong calculus skill.
For x·arctan(x): derivative = arctan x + x/(1+x²). Enter x * arctan(x) to verify. The Implicit Differentiation Calculator handles related cases where arctan appears in implicit equations.
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Key Derivatives to Memorise: d/dx[sin x] = cos x · d/dx[cos x] = −sin x · d/dx[eˣ] = eˣ · d/dx[ln x] = 1/x · d/dx[xⁿ] = nxⁿ⁻¹ · d/dx[arctan x] = 1/(1+x²). Having these six at your fingertips makes the product rule mechanical.
The Product Rule in Advanced Mathematics
The Leibniz Rule for Higher-Order Derivatives
Leibniz Rule — nth Derivative of a Product
(fg)⁽ⁿ⁾ = Σk=0n C(n,k) · f⁽ᵏ⁾ · g⁽ⁿ⁻ᵏ⁾
This formula is structurally identical to the binomial theorem. Use our Higher Order Derivative Calculator to compute nth-order derivatives of product functions directly.
The Product Rule in Multivariable Calculus
For h(x,y) = f(x,y)·g(x,y): ∂h/∂x = (∂f/∂x)·g + f·(∂g/∂x). The gradient satisfies ∇(fg) = g·∇f + f·∇g. Use our Partial Derivative Calculator for multivariable product differentiation.
Integration by Parts — The Inverse of the Product Rule
From (fg)' = f'g + fg', integrating both sides: ∫fg' dx = fg − ∫f'g dx. This is the integration by parts formula. Every time you compute ∫x·eˣ dx, ∫x·ln x dx, or ∫arctan x dx you are applying the product rule in reverse. Mastering the product rule is therefore a direct prerequisite for mastering integration by parts.
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Connection to Higher Mathematics: The algebraic identity d(fg) = f·dg + g·df is the defining property of a derivation in abstract algebra. Any linear map satisfying this Leibniz identity is called a derivation, forming the bridge between calculus and Lie theory.
Product Rule Quick Reference — 12 Common Derivatives
Instant-reference derivatives for the twelve most frequently encountered product rule differentiation combinations. All verified by both engine modes of this calculator.
h(x) = f(x) · g(x)
h'(x) — Product Rule Applied & Simplified
x · sin(x)
sin(x) + x·cos(x)
x² · sin(x)
2x·sin(x) + x²·cos(x)
x · cos(x)
cos(x) − x·sin(x)
x · eˣ
eˣ + x·eˣ = eˣ(x + 1)
x² · eˣ
2x·eˣ + x²·eˣ = xeˣ(x + 2)
eˣ · sin(x)
eˣ·sin(x) + eˣ·cos(x) = eˣ(sin x + cos x)
eˣ · cos(x)
eˣ·cos(x) − eˣ·sin(x) = eˣ(cos x − sin x)
x · ln(x)
ln(x) + 1
x² · ln(x)
2x·ln(x) + x = x(2·ln x + 1)
√x · ln(x)
(ln x + 2) / (2√x)
sin(x) · cos(x)
cos²(x) − sin²(x) = cos(2x)
x · arctan(x)
arctan(x) + x/(1 + x²)
Tips for Faster Product Rule Calculations
Tip 1 — Write (fg)' = f'g + fg' on paper before substituting. Students who skip this template skip terms far more often.
Tip 2 — After applying the product rule, always scan for a common factor. For xeˣ + x²eˣ, factor gives xeˣ(1+x).
Tip 3 — Verify numerically: plug x = 1 into both your derivative and the calculator's result. Matching values confirm correctness.
Tip 4 — For difficult triple-factor products, logarithmic differentiation often produces a cleaner calculation. Use our Implicit Differentiation Calculator for those, and the Taylor Series Calculator when you need series expansions of product functions.
The product rule states that the derivative of a product of two functions f(x) and g(x) is: (f·g)' = f'·g + f·g'. In words: differentiate the first and multiply by the second, then add the first multiplied by the derivative of the second. First published by Leibniz in 1684, it is one of the five fundamental differentiation rules alongside the sum rule, chain rule, quotient rule, and power rule.
Use the product rule when two distinct functions are multiplied together as separate factors: x²·sin(x), eˣ·cos(x). Use the chain rule when one function is nested inside another: sin(x²), e^(cos x). If your expression has both, apply the product rule to the multiplication and the chain rule when differentiating any composite inner factor.
For two functions: (fg)' = f'g + fg'. For three functions: (fgh)' = f'gh + fg'h + fgh'. The Leibniz rule extends to the nth derivative: (fg)⁽ⁿ⁾ = Σ C(n,k)·f⁽ᵏ⁾·g⁽ⁿ⁻ᵏ⁾. In general, the derivative of a product of n functions is the sum of n terms, each formed by differentiating exactly one factor and leaving all others unchanged.
Use * for multiplication, ^ for exponents, and parentheses for grouping. Write log(x) for natural logarithm ln(x), sqrt(x) for square root. Standard trig names: sin, cos, tan, arctan, arcsin, arccos. Examples: x^2 * sin(x), e^x * cos(x), (x+1) * (x^2 - 3), sqrt(x) * log(x), x^3 * e^x * sin(x). Press Enter or click the button to calculate instantly.
🚀 Fast Mode uses a high-performance symbolic computation engine for instant results — ideal for quick verification during study sessions. 🎯 Accurate Mode uses a professional-grade symbolic mathematics system used in academic research, for maximum precision and deeper algebraic simplification. Both modes produce mathematically correct results for all standard calculus expressions. Accurate Mode is especially useful for complex or multi-factor expressions where full simplification depth matters. First use of Accurate Mode loads the computation engine (approximately 10–20 seconds); subsequent calculations in the same session are instant.
Yes. Five export options appear below your result: "Print / PDF" opens your browser's print dialog for a clean PDF. "Save as Text" downloads a formatted plain-text file with all solution steps. "Copy Result" copies the derivative in LaTeX notation. "Share URL" generates a shareable link containing your expression for sharing with classmates or tutors.
Yes. For three functions f, g, h: (fgh)' = f'gh + fg'h + fgh'. Our calculator detects multi-factor products by analysing the expression's syntax tree, so it correctly identifies all factors in expressions like x³·eˣ·sin(x) and applies the full generalised formula automatically.
Writing (fg)' = f'·g' — multiplying the two derivatives. The correct rule always produces two or more terms: (fg)' = f'g + fg'. A related error is forgetting the chain rule on composite factors like sin(x²) or e^(3x).
Write h'(x) = lim[f(x+Δx)g(x+Δx) − f(x)g(x)]/Δx, then add and subtract the bridge term f(x)·g(x+Δx). This splits the limit into two parts that converge to g(x)·f'(x) and f(x)·g'(x) respectively, giving the product rule. The "bridge term" technique is the key non-obvious step.
Integration by parts is derived directly from the product rule. Starting from (fg)' = f'g + fg', integrate both sides: fg = ∫f'g dx + ∫fg' dx. Rearranging: ∫fg' dx = fg − ∫f'g dx. Every time you compute ∫x·eˣ dx, ∫x·ln x dx, or ∫arctan x dx, you are applying the product rule in reverse.
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![Product Rule Formula: d/dx[u(x)·v(x)] = u′(x)·v(x) + u(x)·v′(x) — colour-coded diagram showing factor breakdown](/images/product-rule-formula-diagram.webp)