Formula & Calculator
Chain Rule for Derivatives
Gives the derivative of a composite function by multiplying the derivative of the outer function by the derivative of the inner function.
Interpretation
d/dx[f(g(x))] = f'(g(x))·g'(x). Derivative of a composite function. Allows differentiation of nested functions. Cornerstone of calculus and its applications.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| f'(g(x)) | Derivative of the outer function evaluated at g(x) | |
| g'(x) | Derivative of the inner function |
What it means
The chain rule is the rule for differentiating the composition of two functions. If y = f(g(x)), then the derivative is dy/dx = f'(g(x)) · g'(x). It is often described as "differentiate the outer function, leaving the inner alone, then multiply by the derivative of the inner." The chain rule is essential for differentiating virtually all non‑linear functions, including exponential, logarithmic, trigonometric, and any composite expressions. It is used in physics for time rates of change in complex systems, in economics for elasticity, and in machine learning for backpropagation in neural networks. The chain rule also appears in the integration technique of substitution. Mastering the chain rule is a requirement for success in calculus and its applications in science and engineering.
Worked example
Chain Rule for Derivatives – Two Examples
Real‑World| Parameter | Value |
|---|---|
| f(g) | sin(g) |
| g(x) | x² |
| Parameter | Value |
|---|---|
| f(g) | g³ |
| g(t) | 2t+1 |
Common mistakes
- Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x) – not f'(x) · g'(x).
- Order: Multiply the derivative of outer function (evaluated at inner) by derivative of inner.
- Nested functions: Apply the chain rule repeatedly for multiple compositions.
- Abuse of notation: Do not confuse f'(g(x)) with f'(g(x)) – they mean different things.
- Applications: Essential for differentiating implicit functions, inverse functions, and parametric equations.
Applications
The chain rule, d/dx[f(g(x))] = f'(g(x))·g'(x), is the cornerstone of differentiation for composite functions. It is indispensable in virtually every area of applied mathematics. Engineers use it to differentiate implicit functions, to solve differential equations, and to compute sensitivities in optimisation. In physics, it appears in transformations of variables, such as in coordinate changes. In machine learning, the chain rule is the basis of backpropagation in neural networks. By decomposing complex functions into simpler parts, the chain rule enables tractable derivative computations. Mastering this rule is critical for any professional working with mathematical models, ensuring accurate analysis and robust problem‑solving.
- Backpropagation in deep learning and neural networks
- Coordinate transformations in physics and engineering (e.g., polar to Cartesian)
- Sensitivity analysis and uncertainty quantification
- Implicit differentiation and curve tracing
- Derivation of equations in fluid dynamics and continuum mechanics
Frequently Asked Questions
The chain rule is used to differentiate composite functions: if f(x) = g(h(x)), then f'(x) = g'(h(x)) · h'(x). It says that the derivative of the outer function (evaluated at the inner function) times the derivative of the inner function.
Forgetting to multiply by the inner function's derivative (h'(x)). This is the most common error; people differentiate the outer function but neglect the chain.
Let g(u) = sin u, h(x) = 2x. Then g'(u) = cos u, h'(x) = 2. So f'(x) = cos(2x) · 2 = 2 cos(2x).
If y = f(u) and u = g(x), then dy/dx = dy/du · du/dx. This is a mnemonic that shows the chain rule.
Use the chain rule: f(x) = (sin x)³. Let g(u) = u³, h(x) = sin x. Then f'(x) = 3(sin x)² · cos x.
For f = g(h(k(x))), f' = g'(h(k)) · h'(k) · k'(x).
When differentiating implicitly, treat y as a function of x and apply the chain rule whenever you differentiate a term involving y.
- Differentiating composite functions.
- Implicit differentiation.
- Related rates problems.
- Optimization with constraints.
f(x) = e^(x²). Let g(u) = e^u, h(x) = x². Then f'(x) = e^(x²) · 2x = 2x e^(x²).
The chain rule is the reverse of the substitution method (u‑substitution). When integrating, we look for a function and its derivative to simplify the integral, which is the inverse of the chain rule.