Instantly differentiate any composite function with a full algebraic walkthrough. Supports partial derivatives, multivariable functions, product & quotient rules, nested chain rule, and higher-order derivatives. Powered by a real Computer Algebra System — not a lookup table.
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💡 Use * for multiplication (2*x), ^ for powers (x^2), and function names like sin, cos, exp, ln, sqrt, asin, sinh…
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Product + Chain
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Chain, product & quotient rules
All trig & inverse trig functions
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4+ level deep nested compositions
Higher-order derivatives (1st–10th)
Partial derivatives (multivariable)
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📚 Supported Functions
Trigonometric
sincostancotseccsc
Inverse Trig
asinacosatanacotasecacsc
Hyperbolic
sinhcoshtanhcothsechcsch
Exponential & Log
explnlog
Other
sqrtabspie
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What Is the Chain Rule? A Complete Guide for Students
The chain rule is the differentiation technique used whenever you need to differentiate a composite function — a function where one function is nested inside another. It is one of the five fundamental differentiation rules in calculus, alongside the power rule, product rule, quotient rule, and sum rule.
If you have a function h(x) = f(g(x)), the chain rule states:
The Chain Rule Formula
d/dx[f(g(x))] = f′(g(x)) · g′(x)
In Leibniz notation: dy/dx = (dy/du) · (du/dx), where u = g(x)
In plain language: differentiate the outer function while keeping the inner function unchanged, then multiply by the derivative of the inner function. Think of it as "peeling an onion" — you work from the outermost layer inward, multiplying derivatives at each layer.
Why Does the Chain Rule Work?
The chain rule arises naturally from the definition of the derivative as a limit. When x changes by a small amount Δx, u = g(x) changes by approximately Δu = g′(x)·Δx, and y = f(u) changes by approximately Δy = f′(u)·Δu. Combining these: Δy/Δx ≈ f′(u)·g′(x), which becomes exact in the limit as Δx → 0 (provided g′(x) ≠ 0; the case g′(x) = 0 requires a more careful auxiliary-function argument). This informal reasoning captures the essential intuition behind the result.
Identifying When to Use the Chain Rule
The chain rule is needed whenever you can write your function as h(x) = f(g(x)). Look for these patterns:
A function raised to a power: (x² + 1)⁵ — outer: u⁵, inner: x²+1
A trigonometric function with a non-trivial argument: sin(x²) — outer: sin(u), inner: x²
An exponential with a compound exponent: e^(3x+1) — outer: eᵘ, inner: 3x+1
A logarithm of a compound argument: ln(x³) — outer: ln(u), inner: x³
A square root of an expression: √(x²+4) = (x²+4)^(1/2) — outer: u^(1/2), inner: x²+4
Nested compositions: e^(sin(x²)) — three layers requiring the rule twice
How to Use This Chain Rule Calculator
This tool performs genuine symbolic differentiation — the same mathematical process you would carry out by hand. Here is exactly how to use it:
Enter your function in the input box. Use standard notation: sin(x^2), e^(3*x), ln(x^2+1), (x^2+1)^5, x*sin(x).
Select your variable — choose from 13 options: x, y, t, u, v, s, r, n, z, θ, φ, α, β. The calculator differentiates with respect to the chosen variable and treats all others as constants (partial derivative mode).
Choose the derivative order. Enter 1 for the first derivative, 2 for the second, up to 10.
Choose your Engine: 🚀 Fast Mode gives instant results; 🎯 Accurate Mode uses a full symbolic algebra system for maximum precision on complex expressions.
Choose your detail level: "Full Steps" shows the complete algebraic walkthrough; "Summary" shows key results; "Result Only" gives just the answer.
💡 Input Tip: Always use * for multiplication. Write 2*x, not 2x. Write e^(3*x), not e^3x. Use parentheses generously: sin(x^2) is unambiguous, but sin x^2 may be misread.
Understanding the Output
After calculating, the results panel shows:
Derivative Result — the final simplified answer, rendered in mathematical notation
Input Interpretation — shows how the calculator read your function, so you can verify it was parsed correctly
Function Properties — type, outer function, inner function, complexity, rules used, and domain notes
Alternate Forms — expanded or factored versions of the derivative where available
Numerical Evaluation — the derivative evaluated at x = −1, 0, 0.5, 1, 2 for quick verification
Step-by-Step Solution — a complete algebraic walkthrough of how the derivative was computed
Interactive Graph — plot of f(x) and f′(x) together (loaded on demand)
Chain Rule Worked Examples with Step-by-Step Solutions
The best way to master the chain rule is through carefully worked examples. Below are six examples ranging from basic to advanced, each with a complete step-by-step solution.
Example 1 (Basic): Find d/dx[sin(x²)]
1Identify the structure: This is f(g(x)) where f(u) = sin(u) and g(x) = x².
2Differentiate the outer function (keeping the inner unchanged): f′(g(x)) = cos(x²)
1Identify structure: This is a product u·v where u = x and v = sin(x). Use the Product Rule.
2Product rule: d/dx[u·v] = u′·v + u·v′
3Compute: u′ = 1, v′ = cos(x)
4Result: 1·sin(x) + x·cos(x) = sin(x) + x cos(x)
Chain Rule vs. Product Rule vs. Quotient Rule — Key Differences
Knowing which rule to apply is as important as knowing how to apply it. The three "major" differentiation rules each handle a different structural situation.
Rule
Structure
Formula
Example
Chain Rule
f(g(x)) — composition
f′(g(x)) · g′(x)
sin(x²), e^(3x)
Product Rule
f(x) · g(x) — multiplication
f′g + fg′
x · sin(x)
Quotient Rule
f(x) / g(x) — division
(f′g − fg′) / g²
sin(x) / x
Power Rule
xⁿ — simple power
n · x^(n−1)
x³, x^(1/2)
✅ Quick Decision Test: Ask "What is the last operation performed on x?" If it's a function acting on another function of x → Chain Rule. If it's multiplication of two separate functions of x → Product Rule. If it's division → Quotient Rule.
When You Need More Than One Rule
Many real-world calculus problems require combining rules. For d/dx[x²·sin(x³)]: the outermost operation is multiplication, so start with the product rule. Then the term d/dx[sin(x³)] requires the chain rule. Result: 2x·sin(x³) + x²·cos(x³)·3x² = 2x·sin(x³) + 3x⁴·cos(x³).
Common Chain Rule Mistakes and How to Avoid Them
❌ Mistake 1: Forgetting to Multiply by the Inner Derivative
Wrong: d/dx[sin(x²)] = cos(x²)
Right: d/dx[sin(x²)] = cos(x²) · 2x = 2x cos(x²)
Fix: After differentiating the outer function, always ask yourself "what is the derivative of what's inside?"
❌ Mistake 2: Differentiating the Inner Function First
Wrong order: For e^(x²), differentiating x² first to get 2x, then computing 2x·eˣ²
Right order: Outer first: derivative of eᵘ is eᵘ → e^(x²). Then multiply by inner derivative 2x → 2x·e^(x²)
Fix: Always work outside-in, never inside-out.
❌ Mistake 3: Using Chain Rule on a Simple Polynomial
Unnecessary: d/dx[x³] = 3x² · 1 (technically correct but the ·1 is pointless)
Better: d/dx[x³] = 3x² by the plain power rule
The chain rule's inner derivative is 1 when the inner function is simply x — so it adds nothing.
❌ Mistake 4: Confusing e^(f(x)) with (e^x)^f(x)
Note: e^(x²) means "e raised to the power x²". Its derivative is e^(x²)·2x.
Not: (eˣ)^x² which is a completely different expression requiring logarithmic differentiation.
Chain Rule Quick Reference Table — Common Functions
This table shows the chain rule derivative for the most frequently encountered outer functions. For each, u = g(x) is any differentiable inner function, and u′ = g′(x) is its derivative. Bookmark this as your chain rule "cheat sheet."
Outer Function f(u)
Chain Rule Derivative
Example (u = x²+1)
sin(u)
cos(u) · u′
cos(x²+1) · 2x
cos(u)
−sin(u) · u′
−sin(x²+1) · 2x
tan(u)
sec²(u) · u′
sec²(x²+1) · 2x
eᵘ
eᵘ · u′
e^(x²+1) · 2x
ln(u)
u′ / u
2x / (x²+1)
uⁿ (power)
n · uⁿ⁻¹ · u′
5(x²+1)⁴ · 2x
√u = u^(1/2)
u′ / (2√u)
2x / (2√(x²+1))
asin(u)
u′ / √(1−u²)
2x / √(1−(x²+1)²)
atan(u)
u′ / (1+u²)
2x / (1+(x²+1)²)
sinh(u)
cosh(u) · u′
cosh(x²+1) · 2x
cosh(u)
sinh(u) · u′
sinh(x²+1) · 2x
aᵘ (exponential)
aᵘ · ln(a) · u′
2^(x²+1) · ln(2) · 2x
✅ How to read this table: Find your outer function in column 1. Replace u with your inner function, replace u′ with the inner function's derivative, and multiply. That's the complete chain rule calculation. Need to verify? Enter any of these examples into the calculator above.
The chain rule extends naturally to multivariable calculus. This calculator functions as a partial derivative chain rule calculator — simply select which variable to differentiate with respect to, and all other variables are treated as constants.
How to Compute Partial Derivatives Using This Tool
To find ∂/∂x of a multivariable composite function:
Enter your function (e.g., sin(x^2 + y^2))
Select x as the variable — y will be treated as a constant
Click Calculate to get ∂f/∂x = cos(x²+y²) · 2x
Repeat with y selected to get ∂f/∂y = cos(x²+y²) · 2y
The Multivariable Chain Rule Formula
When z = f(x, y) and both x = x(t), y = y(t) depend on a parameter t, the multivariable chain rule gives:
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
Use this calculator to compute each partial derivative (∂z/∂x and ∂z/∂y) individually, then combine them analytically. For a full guide to multivariable applications, see our Complete Chain Rule Guide.
⚠️ Note on "partial derivative chain rule calculator": Enter functions with multiple variables freely — for example sin(x^2 + y) or exp(x*y). Select x to get ∂/∂x, select y to get ∂/∂y. The tool correctly treats unselected variables as constants during differentiation.
Frequently Asked Questions About the Chain Rule
What is the chain rule in calculus?
The chain rule is a formula for differentiating composite functions — functions where one function is nested inside another. If h(x) = f(g(x)), then h′(x) = f′(g(x)) · g′(x). It is one of the most important rules in differential calculus and is used whenever you encounter expressions like sin(x²), e^(3x+1), or (x²+1)⁵. For a comprehensive explanation with 20+ examples, see our Complete Chain Rule Guide.
Is this calculator accurate — or is it just showing pre-stored answers?
Every result is computed live in your browser using math.js, a production-grade Computer Algebra System (CAS). There are no pre-stored answers. The engine performs genuine symbolic differentiation — the same mathematical process you would carry out by hand — and has been verified against standard calculus references for all supported function types. This is a real mathematical tool, not a demo.
What makes this different from Wolfram Alpha or Symbolab?
Three things: privacy (your expressions are never sent to a server — everything runs in your browser), no paywall on steps (the full step-by-step solution is always free, unlike Wolfram's Pro requirement), and speed (zero network latency since computation is local). For a detailed side-by-side comparison, see our Chain Rule Calculator Guide.
Can I use this for higher-order derivatives?
Yes. Select any order from 1 to 10. The engine differentiates repeatedly and presents the final result with a step-by-step explanation. For example, the 4th derivative of sin(x) returns sin(x) because sin repeats on a 4-cycle of differentiation.
Does the calculator work for the product rule and quotient rule too?
Yes. The engine handles any combination of rules automatically. Enter x*sin(x) and it applies the product rule. Enter sin(x)/x and it applies the quotient rule. Enter x*sin(x^2) and it combines the product rule with the chain rule. You do not need to specify which rule — the engine identifies the structure and applies the appropriate rules. Try our dedicated Product Rule Calculator or Quotient Rule Calculator for focused practice on those rules.
How does the chain rule apply to sin, cos, and other trig functions?
For any trig function with a composite argument: d/dx[sin(f(x))] = cos(f(x)) · f′(x). Similarly, d/dx[cos(f(x))] = −sin(f(x)) · f′(x). Examples: d/dx[sin(x²)] = 2x·cos(x²); d/dx[sin(3x)] = 3cos(3x); d/dx[cos(ln(x))] = −sin(ln(x))/x. The chain rule quick reference table above shows all trig derivatives in one place.
What is the limit chain rule? How does the chain rule relate to limits?
The "limit chain rule" refers to the fact that the chain rule itself is derived from limits. The rigorous proof uses the limit definition of the derivative and the squeeze theorem. It states: if g is differentiable at x and f is differentiable at g(x), then f∘g is differentiable at x and (f∘g)′(x) = f′(g(x))·g′(x). This calculator computes the symbolic result that the limit produces.
What input syntax should I use?
Use * for multiplication, ^ for powers, and function names like sin, cos, exp, ln, sqrt. Examples: sin(x^2), e^(3*x+1), ln(x^2+1), sqrt(x^2+4), x*sin(x). Always include parentheses around function arguments. For a full syntax guide, see the calculator guide.
How do I differentiate with respect to θ (theta)?
Click the θ button in the variable selector, then type θ directly in your expression (you can copy-paste it). Example: type sin(θ^2) with θ selected as the variable. The calculator will correctly compute d/dθ[sin(θ²)] = 2θ cos(θ²).
Is the chain rule used in real life?
Extensively. Engineers use it to model how vibrations propagate through systems. Physicists use it in quantum mechanics and thermodynamics. Economists use it for marginal analysis when cost depends on quantity, which depends on time. In machine learning, the chain rule is the mathematical foundation of backpropagation — the algorithm that trains every modern neural network. Without the chain rule, AI as we know it would not exist.
¿Qué es la regla de la cadena y cómo funciona? (Spanish)
La regla de la cadena (chain rule) es una fórmula para derivar funciones compuestas. Si h(x) = f(g(x)), entonces h′(x) = f′(g(x)) · g′(x). En palabras: se deriva la función exterior manteniendo la interior sin cambios, luego se multiplica por la derivada de la función interior. Esta calculadora aplica la regla de la cadena automáticamente con solución paso a paso.
Limit Chain Rule Calculator — How Limits Underpin the Chain Rule
Every time this calculator produces a derivative, it is computing the result that the limit definition of the chain rule would produce — but symbolically, without approximation. Understanding the limit foundation is essential for calculus courses and for anyone who wants to know why the chain rule is true, not just how to apply it.
The Formal Limit Definition of the Chain Rule
Let h(x) = f(g(x)). Suppose g is differentiable at x = a, and f is differentiable at g(a). The derivative of h at x = a is defined as the following limit:
Chain Rule — Limit Form
h′(a) = limx→a [f(g(x)) − f(g(a))] / (x − a)
This limit equals f′(g(a)) · g′(a) — the chain rule formula.
Why Does the Limit Equal f′(g(a)) · g′(a)?
The key idea is to multiply and divide by [g(x) − g(a)] — but only when g(x) ≠ g(a):
As x → a: the first factor approaches f′(g(a)) (derivative of f at g(a)), and the second factor approaches g′(a) (derivative of g at a). The product of the two limits gives f′(g(a)) · g′(a).
⚠️ The Subtle Gap: The argument above requires g(x) ≠ g(a) near x = a. If g′(a) = 0, then g(x) = g(a) for x near a, and dividing by [g(x) − g(a)] is undefined. The rigorous proof handles this using an auxiliary function φ defined as f′(g(a)) when g(x) = g(a), which removes the division-by-zero problem. This is why textbooks (Apostol, Spivak) present the auxiliary-function proof rather than the informal limit argument.
Worked Example: The Chain Rule via Limits
Let h(x) = sin(x²). We verify h′(x) = 2x cos(x²) using the limit definition:
1Write the limit: h′(a) = limx→a [sin(x²) − sin(a²)] / (x − a)
2Factor: Multiply and divide by (x² − a²): [sin(x²) − sin(a²)] / (x² − a²) · (x² − a²) / (x − a)
3Take the limit of each factor: First factor → cos(a²) [derivative of sin at a²]; second factor → 2a [derivative of x² at a]
4Combine: h′(a) = cos(a²) · 2a = 2a cos(a²) ✓
What Does "Limit Chain Rule Calculator" Mean?
When students search for a limit chain rule calculator, they are typically looking for one of two things: (1) a tool that can differentiate composite functions — which is exactly what this calculator does, computing the symbolic result of the chain rule limit — or (2) a calculator that can evaluate limits of composite functions using the chain rule for limits (which states: if f is continuous at L and limx→a g(x) = L, then limx→a f(g(x)) = f(L)). This calculator addresses use case (1) fully. For limits of composite functions, see our Limit Calculator.
💡 Chain Rule for Limits: The continuity-based chain rule for limits states: limx→a f(g(x)) = f(limx→a g(x)) = f(L), provided f is continuous at L = limx→a g(x). This is used to evaluate limits like limx→0 sin(x²)/x² = 1, by substituting u = x² and recognising that sin(u)/u → 1 as u → 0.
About This Chain Rule Calculator
Verified by Mathematics Experts. This tool was built and is maintained by the DerivativeCalculus.com Editorial Team — educators and mathematicians dedicated to making calculus accessible to every student. All derivative results are produced by the math.js open-source Computer Algebra System (CAS), a battle-tested engine trusted by millions of developers and educators worldwide.
How we ensure accuracy: Every supported function type (trigonometric, inverse trigonometric, hyperbolic, exponential, logarithmic, polynomial, rational) has been verified against standard calculus textbooks including Stewart's Calculus, Apostol's Mathematical Analysis, and Spivak's Calculus. Our Editorial Policy requires that all mathematical claims be independently verifiable by standard CAS tools.
Privacy guarantee: Your browser computes every derivative locally. No function you enter ever leaves your device. We log page visits for analytics (Google Analytics) but never the content of your calculations.
✅CAS Engine: math.js v11.11
✅Last Review: March 18, 2026
✅Student Rating: ⭐ 4.9/5 avg (local votes)
✅Privacy: 100% client-side
Related Derivative Calculators & Resources
Expand your calculus toolkit with these free tools and deep-dive guides from DerivativeCalculus.com. Each resource is designed to complement the chain rule — from foundational concepts to advanced applications.
Antiderivatives and definite integrals — the reverse process of differentiation.
💡 Learning Path: New to the chain rule? Start with our Complete Chain Rule Guide, then practice with the Chain Rule Worksheet. When you need to verify homework answers or explore complex functions, come back to this calculator. For a full walkthrough of how to use this and similar tools, read our Chain Rule Calculator Guide.