Chain Rule Explained: How to Differentiate Composite Functions

Take sin(x²). The derivative of sin is cos, and the power rule handles x². Neither one alone handles the pair. None of the basic rules works on functions like this, even though such functions are closer to the ones you actually meet than the simple kind¹. The power rule alone simply will not get the derivative¹. The chain rule is the missing piece.

What is a composite function?

A composite function is a function inside a function: you feed x into one rule, then feed the result into another. Back in the algebra of functions this is written f(g(x)). The chain rule is the rule for finding the derivative of the composition of two or more functions². Breaking a function into simpler parts that we can already differentiate can get cumbersome, so the chain rule replaces that detour².

Two names do all the work. We identify the inside function and the outside function¹. The outside function is the last operation that we would perform in an evaluation¹. Here is how that sorts out a few functions:

Identifying the outside function is usually simple because it really is the outside function in some sense¹. When it is not obvious, ask which operation you would do last when plugging in a number¹.

The chain rule formula

In words: differentiate the outside function leaving the inside function alone, then multiply all of this by the derivative of the inside function¹. In symbols, with outside function f and inside function g:

Lagrange form: d/dx f(g(x)) = f′(g(x)) · g′(x)³

Leibniz form: if y = f(u) and u = g(x), then dy/dx = dy/du · du/dx³

The 2 forms are the same rule, and the common notation of the chain rule is due to Leibniz³. Each form has its uses, but the first is the one most course work uses¹. When using the Leibniz form, the final answer must be expressed entirely in terms of the original variable given in the problem².

How to interpret the chain rule: rates multiply

Intuitively, knowing the instantaneous rate of change of z relative to y and that of y relative to x allows you to calculate the instantaneous rate of change of z relative to x as the product of the 2 rates of change³. That is the whole meaning. A change in x causes a change in y, which causes a change in z, and each link in the chain contributes one factor³.

Walk through the diagram above. A bicycle is 4 times as fast as a walker, and a car is 2 times as fast as the bicycle, so the car is 4 × 2 = 8 times as fast as the walker. The speeds in the picture are illustrative, and the logic is the one the encyclopedia article uses to explain the rule³. Replace “times as fast” with “rate of change” and you have the chain rule.

The MIT lecture reaches the same place from the algebra. It writes the ratio of changes as a product, the change in y over the change in x times the change in x over the change in t, and then lets the changes shrink⁴. That step works because Δx → 0 too as Δt → 0, because of continuity⁴.

Differentiating a chain means multiplying the links of the chain.

Step by step with worked examples

The method has 4 steps. First, identify the outside and inside functions¹. Second, differentiate the outside function while leaving the inside function alone¹. Third, multiply by the derivative of the inside function¹. Fourth, simplify. When working from the outside in, differentiate one layer at a time².

Worked example (our arithmetic) 1: sin(3x² + x), a function from the course notes¹:

Worked example (our arithmetic) 2: (2x + 1)⁵, with a numeric check by a tiny step²:

Worked example (our arithmetic) 3: e^(−3x), the exponential with the inside left alone¹. Remember that the derivative of the exponential function with the inside left alone is just the original function¹:

Worked example (our arithmetic) 4: ln(x² + 1), a natural logarithm outside an inside¹:

Three layers deep

Sometimes there are more than 2 layers. The derivative of the composition of three functions has three parts, and in general each added function adds a part². Differentiating the composition of three or more functions means applying the chain rule more than once, and we can always work from the outside in, one derivative at a time². There is no formula to memorize: simply apply the chain rule multiple times². In many functions the rule is used more than once, so do not be alarmed when it happens¹. In integration, the counterpart to the chain rule is the substitution rule³.

Worked example (our arithmetic): sin(√(x² + 1)), 3 layers²:

Using the chain rule with other rules

Having the chain rule does not mean the product and quotient rules are no longer needed¹. Some problems are product or quotient rule problems that involve the chain rule¹. Other problems first require the chain rule, and in the process of doing that you need the product or quotient rule¹. The order varies from problem to problem¹. For a product of two functions, first apply the product rule, then apply the chain rule to each term of the product². For a rational expression, the first step is the quotient rule¹. The chain rule can also be used to derive some well-known differentiation rules: the quotient rule is a consequence of the chain rule and the product rule³.

Worked example (our arithmetic): x² sin(3x), first a product, then a chain on the sine¹²:

For the full table of rules to combine with this one, keep the derivative rules cheat sheet open.

What is the chain rule used for in real life?

Any time one changing quantity depends on another that is itself changing, rates multiply. The Leibniz form is used heavily in physics applications². The idea even underlies the backpropagation algorithm, which is used in gradient descent of neural networks in deep learning³.

Worked example (our arithmetic): a spherical balloon. Its volume is V = (4/3)πr³ and its radius grows with time. If the radius is 5 cm and growing at 0.2 cm per second, how fast is the volume growing? The chain rule links the 2 rates, dV/dt = dV/dr · dr/dt³:

A cost example is simpler. If making x units costs C = 50 + 2x dollars, and production is x = 10t units after t weeks, then dC/dt = dC/dx · dx/dt = 2 × 10 = 20 dollars per week³. In every case, the answer must be expressed in terms of the variable the problem asks about².

The chain rule seems to have first been used by Gottfried Wilhelm Leibniz³. He first mentioned it in a 1676 memoir, with a sign error in the calculation, and the notation we still use is his³.

Common mistakes

• Multiplying the whole thing by an inside derivative. One of the more common mistakes is to multiply the whole expression by the “−9” instead of just the term it belongs to¹.
• Differentiating the inside along with the outside. Leave the inside function alone when differentiating the outside function¹.
• Evaluating a derivative at a derivative. We never do that².
• Expecting a single chain rule per problem. There can be 2 terms that each need a separate application¹.
• Choosing the wrong outside function. The outside function is the last operation in an evaluation¹.
• Skipping steps while learning. It is usually best to be careful and write out a couple of extra steps rather than do it all in your head¹. Once you get better, you can do these fairly quickly in your head¹.

Where to go next

The chain rule is one tool in a set. Start with what a derivative is if the idea of a rate of change is still fresh, and read the Fundamental Theorem of Calculus to see the chain rule appear in the first part of that theorem. The core ideas section collects the rest. For the source texts, see Lamar University on the chain rule, OpenStax section 3.6 and the MIT chain rule lecture.