Source: Advanced Calculus I, University of Central Florida
Tags: chain rule, composite functions, derivatives, differentiation, outer function, inner function, function composition, calculus I, UCF
Difficulty: Intermediate Prerequisites: Basic derivative rules (power rule, trigonometric derivatives), understanding of function notation and composition.
The chain rule is one of the core differentiation techniques you will use throughout calculus and beyond. It tells you how to differentiate composite functions, which are functions built by plugging one function into another. Nearly every interesting derivative you encounter in later coursework (multivariable calculus, differential equations, physics) will require the chain rule at some point. You should already be comfortable with basic derivatives (power rule, trig functions) and understand what f(g(x)) means before working through these notes.
The chain rule lets you differentiate a composite function by taking the derivative of the outer function (evaluated at the inner function) and multiplying by the derivative of the inner function. In short: differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.
Composite function
A function formed by plugging one function into another. If g(x) = sin(x) and h(x) = x², then f(x) = g(h(x)) = sin(x²) is a composite function. In simple terms, it is a "function of a function."
Outer function
The function that wraps around the other. In sin(x²), the outer function is sin(·). Think of it as the last operation you would perform if you were evaluating the expression step by step.
Inner function
The function that sits inside the other. In sin(x²), the inner function is x². Think of it as the first operation you would perform when evaluating.
Chain rule
The differentiation rule for composite functions. If f(x) = g(h(x)), then f'(x) = g'(h(x)) · h'(x). In simple terms, this means: "derivative of the outside, evaluated at the inside, times the derivative of the inside."
Before applying the chain rule, you need to spot that a function is composite.
Ask yourself: "Is there a function inside another function?" If yes, you have a composite.
Break it into outer and inner parts:
f(x) = sin(x²): outer = sin(·), inner = x²
f(x) = (3x + 1)⁵: outer = (·)⁵, inner = 3x + 1
f(x) = e^(2x): outer = e^(·), inner = 2x
Step 1: Identify the outer function g and the inner function h.
Step 2: Differentiate the outer function, but leave the inner function in place: g'(h(x)).
Step 3: Multiply by the derivative of the inner function: h'(x).
Step 4: Simplify.
Outer function: g(x) = sin(x), so g'(x) = cos(x)
Inner function: h(x) = x², so h'(x) = 2x
Apply the chain rule:
f'(x) = g'(h(x)) · h'(x)
f'(x) = cos(x²) · 2x
f'(x) = 2x cos(x²)
When functions are nested more than two layers deep, apply the chain rule repeatedly from the outside in.
Example: f(x) = sin((2x + 1)³)
Outermost: sin(·), next layer: (·)³, innermost: 2x + 1
f'(x) = cos((2x + 1)³) · 3(2x + 1)² · 2 = 6(2x + 1)² cos((2x + 1)³)
Chain rule (Lagrange notation):
If f(x) = g(h(x)), then f'(x) = g'(h(x)) · h'(x)
Chain rule (Leibniz notation):
If y = g(u) and u = h(x), then dy/dx = (dy/du) · (du/dx)
The Leibniz form is especially useful for related rates and implicit differentiation, and it makes the "multiply the derivatives" logic feel intuitive, almost like cancelling fractions (though that is not literally what is happening).
The chain rule appears any time a quantity depends on another quantity that itself depends on something else. For instance, if the temperature of a metal rod depends on its distance from a heat source, and that distance changes with time, the chain rule tells you how the temperature changes with time. In machine learning, backpropagation (the algorithm that trains neural networks) is essentially the chain rule applied across many layers of composed functions.
Forgetting to multiply by the inner derivative. The most common error. Students differentiate sin(x²) and write cos(x²), dropping the · 2x. The inner derivative is never 1 unless the inner function is just x.
Confusing the product rule with the chain rule. sin(x) · x² needs the product rule. sin(x²) needs the chain rule. The difference is whether two functions are multiplied together or nested inside each other.
Applying the chain rule when it is not needed. A function like f(x) = x² + sin(x) is a sum, not a composition. Use the sum rule, not the chain rule.
Stopping too early with nested compositions. For something like e^(sin(x²)), you need to chain three times. Each layer requires its own derivative factor.
⚠️ The chain rule is tested on virtually every calculus exam. Expect to use it in combination with the product rule, quotient rule, or both.
⚠️ Exam problems frequently disguise composite functions. Rewriting expressions to spot the composition (e.g., √(3x + 1) = (3x + 1)^(1/2)) is a tested skill.
⚠️ Leibniz notation (dy/dx = dy/du · du/dx) appears in related rates and implicit differentiation problems, so fluency in both notations matters.
True or false: The derivative of cos(5x) is -sin(5x).
A: False. The derivative is -5 sin(5x). You must multiply by the derivative of the inner function 5x.
Fill in the blank: If f(x) = (x³ + 2)⁴, then f'(x) = 4(x³ + 2)³ · ______.
A: 3x²
True or false: The chain rule and the product rule are interchangeable.
A: False. The chain rule is for compositions; the product rule is for products of two functions.
Q: Find the derivative of f(x) = e^(3x²).
A: f'(x) = e^(3x²) · 6x. The outer function is e^(·) and the inner function is 3x². The derivative of e^(·) is itself, and the derivative of 3x² is 6x.
Q: Find the derivative of f(x) = ln(sin(x)).
A: f'(x) = (1/sin(x)) · cos(x) = cos(x)/sin(x) = cot(x). Outer is ln(·), inner is sin(x).
Q: Differentiate f(x) = (2x – 7)^(1/2).
A: f'(x) = (1/2)(2x – 7)^(-1/2) · 2 = 1/√(2x – 7). Rewrite the square root as a power of 1/2, then apply the chain rule.
Q: If y = cos³(x), what is dy/dx?
A: Rewrite as y = (cos(x))³. Then dy/dx = 3(cos(x))² · (-sin(x)) = -3 cos²(x) sin(x).
Q: When should you use the chain rule versus the product rule?
A: Use the chain rule when one function is inside another, i.e. f(g(x)). Use the product rule when two functions are multiplied together, i.e. f(x) · g(x). Some problems require both.
This connects directly to implicit differentiation, because implicit differentiation treats y as a function of x and applies the chain rule every time you differentiate a term containing y. It also connects to related rates, where you differentiate an equation with respect to time using the chain rule on every variable that changes with t. In multivariable calculus, the chain rule generalises to partial derivatives and Jacobian matrices.
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