Chain Rule Calculator

A chain rule calculator finds the derivative of a composite function — a function built by feeding one function into another, like (2x + 1)⁵ or sin(3x). Composite functions appear everywhere in calculus courses and in real models, and differentiating them by hand is where most sign errors and dropped factors happen, so this tool exists to give a reliable result you can check your working against.

Quick Answer

If y = f(g(x)), then dy/dx = f′(g(x)) × g′(x) — differentiate the outer function while leaving the inner one untouched, then multiply by the derivative of the inner function. For y = (2x + 1)⁵ the result is 10(2x + 1)⁴, and at x = 1 that equals 810.

Why the Extra Multiplication Appears

Think of a composite function as two machines in sequence: x goes in, the inner function g transforms it, and the outer function f transforms the result. If the inner quantity is changing twice as fast as x, the final output changes twice as fast too — the rates multiply. That is all the chain rule says. In Leibniz notation it reads dy/dx = (dy/du) × (du/dx), where u is the inner function; the du terms appear to cancel, which is a handy memory aid even though the rigorous justification is subtler.

The method is always the same three moves: identify the inner function (the part you would compute first if you were evaluating the expression at a number), apply the ordinary derivative rule to the outer function while treating the inner part as a single lump, and then multiply by the derivative of that lump. Powers, roots, exponentials, logarithms, and trig functions wrapped around any polynomial are all handled by this one pattern, and genuinely nested functions — sin(e^(2x)), say — simply apply the pattern once per layer, working from the outside in.

How to Use the Chain Rule Calculator

1. Write out the function you need to differentiate and confirm it really is composite — one function inside another.
2. Enter the function into the calculator exactly as written, using brackets to make the grouping unambiguous.
3. Read the derivative it returns.
4. If you need the slope at a specific point, substitute that x-value into the derivative.
5. Compare with your own working step by step: name the inner function, differentiate the outer, and check whether the final multiplication is present.
6. For triple-layer functions, peel one layer at a time rather than trying to jump straight to the answer.

Worked Example: Raising a Bracket to the Fifth Power

Differentiate f(x) = (2x + 1)⁵. The inner function is u = 2x + 1 and the outer function is u⁵. Differentiating the outer while keeping the lump intact gives 5u⁴, and the derivative of the inner is 2. Multiplying: f′(x) = 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴. At x = 1 the bracket equals 3, so f′(1) = 10 × 3⁴ = 10 × 81 = 810 — the curve is climbing very steeply there, which makes sense for a fifth power. The same pattern produces these standard results:

• d/dx [sin(3x)] = 3cos(3x)
• d/dx [e^(4x)] = 4e^(4x)
• d/dx [(x² + 1)³] = 6x(x² + 1)²
• d/dx [√(5x)] = 5 ÷ (2√(5x))
• d/dx [ln(sin x)] = cos x ÷ sin x = cot x

When two layers are comfortable, try three, because the method does not change — it just repeats. Differentiate y = sin(e^(2x)). The outermost layer is sine, whose derivative is cosine with the inside untouched: cos(e^(2x)). The next layer is e raised to a power, which reproduces itself but is itself composite, contributing a factor of e^(2x). The innermost layer, 2x, contributes the final factor of 2. Chained together: dy/dx = 2e^(2x) · cos(e^(2x)). Notice the order of the working — outside in, one factor per layer, nothing skipped. If a hand answer and the calculator disagree on a problem like this, count the factors in each: the number of factors should equal the number of layers, and a missing one names the layer that was merged into its neighbour by mistake.

The chain rule also quietly underlies the "related rates" problems that end most introductory courses. If a balloon's radius grows at 0.5 cm per second, the volume V = (4/3)πr³ changes at dV/dt = 4πr² × dr/dt — the derivative of volume with respect to radius, multiplied by the derivative of radius with respect to time. That multiplication by the "inner rate" is the chain rule wearing a wristwatch, and recognising it turns a feared problem type into a familiar one.

The Mistakes That Cost the Most Marks

One error accounts for most lost marks in this topic: differentiating the outer function and forgetting to multiply by the derivative of the inner one. Writing 5(2x + 1)⁴ as the derivative of (2x + 1)⁵ is wrong by a factor of 2 everywhere, and the error is invisible unless you check. The second is treating a composite as if it were a product — (2x + 1)⁵ is not 2x + 1 times something, so the product rule does not apply until the bracket is expanded. Third, order of operations when evaluating: substitute into the derivative after simplifying, and raise the whole bracket to its power before multiplying. Fourth, with trigonometric inner functions, watch the sign: d/dx [cos(2x)] = −2sin(2x), and the negative comes from the outer rule, not the inner. Finally, when the inner function is itself a product or quotient, its derivative needs its own rule first — the chain rule does not replace those, it wraps around them.

Where the Chain Rule Leads Next

Mastering this pattern pays off across the rest of differential calculus. The general Derivative Calculator applies the chain rule together with every other basic rule, and the Product Rule Calculator covers the partner case where two functions are multiplied rather than composed — learning to tell those two structures apart at a glance is most of first-term calculus. Both tools, and the rest of the differentiation set, are collected under Math Calculators.