info@scisolvelab.com
Icon-facebook Twitter Icon-linkedin
/

Product Rule Calculator — Derivative of a Product with Steps

Product Rule Calculator — Derivative of a Product with Steps
Calculus Tool

Product Rule Calculator

Apply the product rule d/dx[u·v] = u'v + uv' to any product of functions — handles two-factor products, triple products, product+chain rule combinations, and partial derivative product rule — with full step-by-step working and color-coded u/v identification throughout every display.

d/dx[ u · v ] = u' · v + u · v'
Product Rule Calculator — d/dx[u·v] = u'v + uv'

Use * for multiplication, ^ for powers. Auto-detects u and v by splitting on *.

d/dx[ u · v ] = ?
Variable:
x²·sin(x)
eˣ·ln(x)
x³·cos(x)
√x·eˣ
(x²+1)·tan(x)
x·ln(x)
x·sin(x)
x²·eˣ
Error

Product Rule Result

—
Step-by-Step Working

Differentiate a product of three functions using d/dx[u·v·w] = u'vw + uv'w + uvw'

u =
v =
w =
Variable:
x²·sin(x)·eˣ
x·cos(x)·ln(x)
x³·eˣ·sin(x)
Error

Triple Product Rule Result

—
Step-by-Step Working

When one or both factors are composite functions, the chain rule must be applied to differentiate each factor before assembling with the product rule.

u =
v =
Variable:
sin(x²)·e^(3x)
(x²+1)⁵·ln(2x)
cos(x³)·x²eˣ
Error

Product + Chain Rule Result

—
Step-by-Step Working

Partial derivative product rule: ∂/∂x[u·v] = (∂u/∂x)·v + u·(∂v/∂x). All other variables are treated as constants.

u =
v =
Differentiate with respect to:
x²y · sin(xy)
eˣy² · cos(xy)
xy · ln(xy)
Error

Partial Derivative Product Rule Result

—
Step-by-Step Working

The Product Rule Formula — d/dx[uv] = u'v + uv'

The product rule calculator above applies the product rule d/dx[u·v] = u'v + uv' automatically — handling two-factor products, triple products, product-chain rule combinations, and partial derivative product rule with complete step-by-step working and color-coded u/v identification. Product rule differentiation is one of the four fundamental derivative rules in calculus.

The product rule states: if h(x) = u(x) · v(x), then h'(x) = u'(x)·v(x) + u(x)·v'(x). In plain English: derivative of the first times the second, plus the first times the derivative of the second.

d/dx[u·v] = u'·v + u·v' The product rule formula — derivative of first × second + first × derivative of second

All three notation forms are equivalent — memorise all of them as your product rule solver may present any one:

(uv)' = u'v + uv'
Prime notation
d/dx[u·v] = u'v + uv'
Leibniz notation
d/dx[f·g] = f'g + fg'
Function notation

Memory trick: "Derivative of first times second, plus first times derivative of second." The color coding in our product rule calculator makes this concrete — indigo = u (first factor), teal = v (second factor). Track the colors across u'v + uv' and you never lose which term is which.

When to use the product rule: Any time two functions are multiplied together — not added, subtracted, or composed. d/dx[u + v] uses the sum rule. d/dx[f(g(x))] uses the chain rule. d/dx[u · v] uses the product rule — d/dx[uv] = u'v + uv'.

How to Use the Product Rule — Step-by-Step Method

The four-step method for applying the product rule to any product of functions:

  1. Step 1 — Identify u and v: Label the first factor u(x) and the second factor v(x). Any split works, but choosing the simpler function as u often simplifies the calculation.
  2. Step 2 — Differentiate u: Find u'(x) using the appropriate rule (power rule, trig rule, exponential rule, etc.).
  3. Step 3 — Differentiate v: Find v'(x) using the appropriate rule. If v is composite, apply the chain rule here.
  4. Step 4 — Substitute into d/dx[uv] = u'v + uv': Write out both terms, simplify, and combine like terms.

Example 1 — Polynomial × Trig: d/dx[x²·sin(x)]

  1. u = x², v = sin(x)
  2. u' = 2x (power rule)
  3. v' = cos(x) (trig rule)
  4. d/dx[u·v] = u'v + uv' = (2x)(sin x) + (x²)(cos x)
  5. Answer: 2x·sin(x) + x²·cos(x)

Example 2 — Exponential × Logarithm: d/dx[eˣ·ln(x)]

  1. u = eˣ, v = ln(x)
  2. u' = eˣ (exponential rule — eˣ is its own derivative)
  3. v' = 1/x (log rule)
  4. d/dx[uv] = u'v + uv' = eˣ·ln(x) + eˣ·(1/x)
  5. Answer: eˣ·ln(x) + eˣ/x = eˣ(ln(x) + 1/x)

Example 3 — Polynomial × Exponential: d/dx[x³·eˣ]

  1. u = x³, v = eˣ
  2. u' = 3x², v' = eˣ
  3. d/dx[uv] = u'v + uv' = 3x²·eˣ + x³·eˣ = eˣ(3x² + x³) = x²eˣ(3+x)
  4. Answer: x²eˣ(x+3)

Example 4 — Radical × Trig: d/dx[√x·cos(x)]

  1. u = √x = x^(1/2), v = cos(x)
  2. u' = (1/2)x^(−1/2) = 1/(2√x), v' = −sin(x)
  3. d/dx[uv] = u'v + uv' = cos(x)/(2√x) + √x·(−sin(x))
  4. Answer: cos(x)/(2√x) − √x·sin(x)

Example 5 — Product Rule vs Expanding: d/dx[(x+1)(x+2)]

Method A — Product Rule:

  1. u = x+1, v = x+2; u' = 1, v' = 1
  2. d/dx[uv] = 1·(x+2) + (x+1)·1 = x+2+x+1 = 2x+3

Method B — Expand first:

  1. (x+1)(x+2) = x²+3x+2
  2. d/dx[x²+3x+2] = 2x+3 ✓ — same answer confirms both methods are correct

Product Rule Practice Problems — 8 Solved Examples

Eight fully worked product rule practice problems of increasing difficulty. Click each to expand the full solution. These cover every function type you will encounter when using the product rule to find derivative values.

1 d/dx[x · sin(x)] = ?
  1. Identify: u = x, v = sin(x)
  2. Differentiate: u' = 1 (constant rule), v' = cos(x) (trig rule)
  3. Apply d/dx[uv] = u'v + uv': 1·sin(x) + x·cos(x)
  4. Answer: sin(x) + x·cos(x)
  5. Verify: check at x=1 numerically ✓
2 d/dx[x² · eˣ] = ?
  1. Identify: u = x², v = eˣ
  2. Differentiate: u' = 2x (power rule), v' = eˣ (exponential rule)
  3. Apply d/dx[uv] = u'v + uv': 2x·eˣ + x²·eˣ
  4. Factor: eˣ(2x + x²) = xeˣ(2 + x)
  5. Answer: xeˣ(x + 2)
3 d/dx[x · ln(x)] = ? (classic result)
  1. Identify: u = x, v = ln(x)
  2. Differentiate: u' = 1, v' = 1/x (log rule)
  3. Apply d/dx[uv] = u'v + uv': 1·ln(x) + x·(1/x) = ln(x) + 1
  4. Note: x·(1/x) = 1 simplifies beautifully
  5. Answer: ln(x) + 1 — the classic product rule result
4 d/dx[√x · cos(x)] = ?
  1. Rewrite: √x = x^(1/2)
  2. Identify: u = x^(1/2), v = cos(x)
  3. Differentiate: u' = (1/2)x^(−1/2) = 1/(2√x), v' = −sin(x)
  4. Apply d/dx[uv] = u'v + uv': cos(x)/(2√x) + x^(1/2)·(−sin(x))
  5. Answer: cos(x)/(2√x) − √x·sin(x)
5 d/dx[eˣ · sin(x)] = ?
  1. Identify: u = eˣ, v = sin(x)
  2. Differentiate: u' = eˣ, v' = cos(x)
  3. Apply d/dx[uv] = u'v + uv': eˣ·sin(x) + eˣ·cos(x)
  4. Factor: eˣ(sin(x) + cos(x))
  5. Answer: eˣ(sin(x) + cos(x))
  6. Numerical check at x=1.5: eˣ(sin+cos) ≈ 4.482×(0.997+0.071) ≈ 4.787 ✓
6 d/dx[(x²+1) · (x³−3x)] = ?
  1. Identify: u = x²+1, v = x³−3x
  2. Differentiate: u' = 2x, v' = 3x²−3
  3. Apply d/dx[uv] = u'v + uv': 2x(x³−3x) + (x²+1)(3x²−3)
  4. Expand term 1: 2x⁴ − 6x²
  5. Expand term 2: 3x⁴ − 3x² + 3x² − 3 = 3x⁴ − 3
  6. Combine: 2x⁴ − 6x² + 3x⁴ − 3 = 5x⁴ − 6x² − 3
  7. Answer: 5x⁴ − 6x² − 3
7 d/dx[x³ · sin(x) · eˣ] = ? (triple product)
  1. Triple product rule: d/dx[uvw] = u'vw + uv'w + uvw'
  2. u = x³, v = sin(x), w = eˣ
  3. u' = 3x², v' = cos(x), w' = eˣ
  4. Term 1: 3x²·sin(x)·eˣ
  5. Term 2: x³·cos(x)·eˣ
  6. Term 3: x³·sin(x)·eˣ
  7. Combine: eˣ(3x²sin(x) + x³cos(x) + x³sin(x))
  8. Answer: eˣ(3x²·sin(x) + x³·cos(x) + x³·sin(x))
8 d/dx[sin(x²) · e^(3x)] = ? (product + chain)
  1. Both factors are composite — chain rule needed for each
  2. u = sin(x²) — outer: sin(·), inner: x² → u' = cos(x²)·2x = 2x·cos(x²)
  3. v = e^(3x) — outer: eᵘ, inner: 3x → v' = e^(3x)·3 = 3e^(3x)
  4. Apply d/dx[uv] = u'v + uv': (2x·cos(x²))(e^(3x)) + (sin(x²))(3e^(3x))
  5. Factor e^(3x): e^(3x)(2x·cos(x²) + 3·sin(x²))
  6. Answer: e^(3x)(2x·cos(x²) + 3·sin(x²))

Product Rule vs Chain Rule — When to Use Each

The most common confusion in calculus — knowing which rule to apply and when. The product rule and chain rule are both essential, but they solve entirely different problems.

Decision Guide — Which Rule?

Are two functions multiplied?
e.g. x² · sin(x)
→
Product Rule
d/dx[uv] = u'v + uv'
Is one function inside another?
e.g. sin(x²)
→
Chain Rule
d/dx[f(g(x))] = f'(g(x))·g'(x)
Is a composite function multiplied by another?
e.g. sin(x²) · eˣ
→
Both Rules
Chain rule inside, product rule outside

Critical distinction: f(x)·g(x) (product — use product rule d/dx[uv]=u'v+uv') vs f(g(x)) (composition — use chain rule). The most common mistake is treating a product as a composition or vice versa. These are completely different operations with completely different rules.

Product vs Composition — Concrete Examples

  • x² · sin(x) — PRODUCT → product rule → 2x·sin(x) + x²·cos(x)
  • sin(x²) — COMPOSITION → chain rule → cos(x²)·2x
  • x² · sin(x²) — PRODUCT where second factor is composite → both rules: product rule outer, chain rule to get derivative of sin(x²)

Triple Product Rule — Differentiating Three Functions

When three functions are multiplied, apply the product rule twice. The result is the triple product rule, and the pattern is elegant: each term has exactly one differentiated factor.

d/dx[u·v·w] = u'vw + uv'w + uvw' Each term has exactly one primed (differentiated) factor — three terms total

Derivation: Group (u·v) as one function, apply the product rule to [(uv)·w]:

  1. d/dx[(uv)·w] = (uv)'·w + (uv)·w'
  2. Expand (uv)' using the product rule: (uv)' = u'v + uv'
  3. Substitute: (u'v + uv')w + uvw' = u'vw + uv'w + uvw' ✓

Pattern: For n factors, there are n terms — each with exactly one factor differentiated and n−1 factors undifferentiated. This generalises to the product rule for any number of functions.

Triple Product Example: d/dx[x² · sin(x) · eˣ]

  1. u = x², v = sin(x), w = eˣ
  2. u' = 2x, v' = cos(x), w' = eˣ
  3. Term 1 (u'vw): 2x·sin(x)·eˣ
  4. Term 2 (uv'w): x²·cos(x)·eˣ
  5. Term 3 (uvw'): x²·sin(x)·eˣ
  6. Combine: eˣ(2x·sin(x) + x²·cos(x) + x²·sin(x))

Product Rule with Chain Rule — Composite Factors

When u or v is a composite function, apply the chain rule to find its derivative first, then plug into the product rule d/dx[uv] = u'v + uv'. The order is always: chain rule inside → product rule outside.

Three-step method for composite factor product rule:

  1. Identify which factor(s) are composite — f(g(x)) form where inner function ≠ plain variable
  2. Apply chain rule to each composite factor: d/dx[f(g(x))] = f'(g(x))·g'(x)
  3. Plug the derivatives into d/dx[uv] = u'v + uv' — the product rule formula

Example 1: d/dx[sin(x²) · e^(3x)]

  1. Factor 1 u = sin(x²) — composite, requires chain rule
  2. Chain rule: outer = sin(·), inner = x² → u' = cos(x²)·2x = 2x·cos(x²)
  3. Factor 2 v = e^(3x) — composite, requires chain rule
  4. Chain rule: outer = eᵘ, inner = 3x → v' = e^(3x)·3 = 3e^(3x)
  5. Apply d/dx[uv] = u'v + uv': (2x·cos(x²))(e^(3x)) + (sin(x²))(3e^(3x))
  6. Answer: e^(3x)(2x·cos(x²) + 3·sin(x²))

Example 2: d/dx[(x²+1)⁵ · ln(2x)]

  1. u = (x²+1)⁵ — composite power → chain rule: u' = 5(x²+1)⁴·2x = 10x(x²+1)⁴
  2. v = ln(2x) — composite log → chain rule: v' = (1/(2x))·2 = 1/x
  3. Apply d/dx[uv] = u'v + uv': 10x(x²+1)⁴·ln(2x) + (x²+1)⁵·(1/x)
  4. Answer: 10x(x²+1)⁴·ln(2x) + (x²+1)⁵/x

Partial Derivative Product Rule

The partial derivative product rule is identical in structure to the standard product rule — only the notation changes. When differentiating ∂/∂x[u(x,y)·v(x,y)], treat all variables other than x as constants throughout.

∂/∂x[u·v] = (∂u/∂x)·v + u·(∂v/∂x) Same formula as d/dx[uv] = u'v + uv' — treat all other variables as constants

Example: ∂/∂x[x²y · sin(xy)]

  1. u = x²y (treat y as constant), v = sin(xy)
  2. ∂u/∂x = 2xy (y is constant, so ∂/∂x[x²y] = 2xy)
  3. ∂v/∂x = cos(xy)·y (chain rule — inner function xy has ∂/∂x = y)
  4. Apply: (2xy)·sin(xy) + (x²y)·y·cos(xy)
  5. Answer: 2xy·sin(xy) + x²y²·cos(xy)

For the full multivariable derivative tool, see our ∂ Partial Derivative Calculator.

Common Mistakes With the Product Rule

Mistake 1 — Multiplying Derivatives: (uv)' ≠ u'·v' (the most common error)

  • ❌ Wrong: d/dx[x²·sin(x)] = 2x·cos(x)
  • ✅ Correct: d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x) — always TWO terms
  • The product rule d/dx[uv] = u'v + uv' gives a SUM, not a product of derivatives

Mistake 2 — Stopping After One Term

  • ❌ Wrong: d/dx[x·ln(x)] = 1·ln(x) = ln(x) — missing the second term
  • ✅ Correct: d/dx[x·ln(x)] = 1·ln(x) + x·(1/x) = ln(x) + 1 — always write BOTH u'v AND uv'

Mistake 3 — Applying Product Rule to Simple Polynomial Products

  • For d/dx[(x+1)(x+2)] — it is faster to expand first: x²+3x+2 → 2x+3
  • The product rule gives the same answer but takes longer for simple polynomial × polynomial cases

Mistake 4 — Sign Errors in v'

  • ❌ Wrong: d/dx[x²·sin(x)] with v' = sin(x) instead of v' = cos(x)
  • ❌ Wrong: d/dx[x·cos(x)] with v' = sin(x) instead of v' = −sin(x)
  • ✅ Always verify trig derivatives: d/dx[sin(x)] = cos(x), d/dx[cos(x)] = −sin(x)

Mistake 5 — Not Applying Chain Rule Within Product Rule

  • ❌ Wrong: d/dx[sin(x²)·eˣ] with u' = cos(x²) — missing the 2x from chain rule
  • ✅ Correct: u = sin(x²) → u' = cos(x²)·2x = 2x·cos(x²) — always check if factor is composite

Additional Worked Examples

Example 6: d/dx[x⁴ · ln(x)]

  1. u = x⁴, v = ln(x); u' = 4x³, v' = 1/x
  2. d/dx[uv] = u'v + uv' = 4x³·ln(x) + x⁴·(1/x) = 4x³·ln(x) + x³
  3. Answer: x³(4·ln(x) + 1)

Example 7: d/dx[(x²+3x) · eˣ]

  1. u = x²+3x, v = eˣ; u' = 2x+3, v' = eˣ
  2. d/dx[uv] = u'v + uv' = (2x+3)eˣ + (x²+3x)eˣ = eˣ(2x+3+x²+3x)
  3. Answer: eˣ(x²+5x+3)

Example 8: d/dx[tan(x) · ln(x)]

  1. u = tan(x), v = ln(x); u' = sec²(x), v' = 1/x
  2. d/dx[uv] = u'v + uv' = sec²(x)·ln(x) + tan(x)/x
  3. Answer: sec²(x)·ln(x) + tan(x)/x

Example 9: d/dx[x·eˣ·sin(x)] (triple product)

  1. u = x, v = eˣ, w = sin(x)
  2. u' = 1, v' = eˣ, w' = cos(x)
  3. d/dx[uvw] = 1·eˣ·sin(x) + x·eˣ·sin(x) + x·eˣ·cos(x)
  4. Answer: eˣ(sin(x) + x·sin(x) + x·cos(x)) = eˣ((1+x)sin(x) + x·cos(x))

Example 10: d/dx[√x · ln(x)] (radical × log)

  1. u = x^(1/2), v = ln(x); u' = 1/(2√x), v' = 1/x
  2. d/dx[uv] = u'v + uv' = ln(x)/(2√x) + √x·(1/x) = ln(x)/(2√x) + 1/√x
  3. Answer: (ln(x) + 2)/(2√x)

Example 11: d/dx[x²·cos(x)·ln(x)] (triple product)

  1. u = x², v = cos(x), w = ln(x)
  2. u' = 2x, v' = −sin(x), w' = 1/x
  3. Term 1: 2x·cos(x)·ln(x)
  4. Term 2: x²·(−sin(x))·ln(x) = −x²·sin(x)·ln(x)
  5. Term 3: x²·cos(x)·(1/x) = x·cos(x)
  6. Answer: 2x·cos(x)·ln(x) − x²·sin(x)·ln(x) + x·cos(x)

Frequently Asked Questions

What is the product rule?
The product rule states that the derivative of a product of two functions is: d/dx[u·v] = u'·v + u·v'. Derivative of the first times the second, plus the first times the derivative of the second. It applies whenever two functions are multiplied — for example x²·sin(x), eˣ·ln(x), or √x·cos(x).
When do you use the product rule?
Use the product rule whenever you differentiate a product of two or more distinct functions — f(x)·g(x). If functions are added, use the sum rule. If one is inside another (f(g(x))), use the chain rule. If a composite function is part of a product, use both the chain rule and the product rule.
What is the formula for the product rule?
The product rule formula: d/dx[u·v] = u'v + uv'. Equivalent forms: (uv)' = u'v + uv' (prime notation), d/dx[fg] = f'g + fg' (function notation). All three mean the same thing. The formula d/dx[uv] = u'v + uv' must be memorised — it is one of the four fundamental derivative rules.
What is the difference between the product rule and the chain rule?
Product rule: d/dx[u·v] = u'v + uv' — for two functions MULTIPLIED. Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x) — for one function INSIDE another. When a composite function appears in a product, use both: chain rule to differentiate the composite factor, then product rule to assemble the result.
What is the triple product rule?
The triple product rule: d/dx[u·v·w] = u'vw + uv'w + uvw'. Each term has exactly one differentiated factor. Derived by applying the product rule twice: treat (uv) as one function, apply product rule to (uv)·w, then expand (uv)' using the product rule again. Pattern: n functions → n terms, each with one primed factor.
How do you apply the product rule to partial derivatives?
Identical formula with ∂ notation: ∂/∂x[u·v] = (∂u/∂x)·v + u·(∂v/∂x). The only difference is treating all variables other than the differentiation variable as constants. For ∂/∂x[x²y·sin(xy)]: treat y as constant, apply the partial derivative product rule exactly as you would apply d/dx[uv] = u'v + uv'.
Can you use the product rule for more than two functions?
Yes. For three functions: d/dx[uvw] = u'vw + uv'w + uvw'. For n functions, there are n terms — each with exactly one factor differentiated and the rest unchanged. Our triple product rule calculator handles three-factor cases automatically with full step-by-step working showing all three terms.

Related Calculators

⛓
Chain Rule Calculator — for composite functions
÷
Quotient Rule Calculator — for f/g derivatives
dy
Implicit Differentiation — product rule applied to y(x)
∂
Partial Derivative Calculator — multivariable product rule
∫
Antiderivative Calculator — reverse of differentiation
Quick Rules
d/dx[u·v] = u'v + uv' Product Rule — core formula
(uv)' = u'v + uv' Prime notation form
d/dx[uvw] = u'vw + uv'w + uvw' Triple product rule
∂/∂x[u·v] = u_x·v + u·v_x Partial derivative form
(uv)' ≠ u'v' — WRONG! Common mistake to avoid
Quick Examples
x²·sin(x) → 2x·sin(x)+x²·cos(x)
eˣ·ln(x) → eˣln(x)+eˣ/x
x·ln(x) → ln(x)+1
x³·cos(x) → 3x²cos(x)−x³sin(x)
x·sin(x) → sin(x)+x·cos(x)
x²·eˣ → xeˣ(x+2)
√x·eˣ → eˣ/(2√x)+eˣ√x
(x²+1)·tan(x)
Related Tools
⛓
Chain Rule Calc
÷
Quotient Rule Calc
∂
Partial Derivative
∫
Antiderivative Calc

Share This Tool

Share the Product Rule Calculator with students!

Free chemistry, physics, biology & math calculators with step-by-step solutions. Trusted by 100,000+ students. Solve any science problem instantly!

Pages

Newsletter

Subscribe to our Newsletter to be updated. We promise not to spam.

Copyright © 2026 SciSolveLab. All Rights Reserved

Icon-facebook Instagram X-twitter Icon-linkedin
Scroll to Top