复合函数 · 每一层都看得见

带详细步骤的链式法则计算器

计算 (x²+1)³、sin(x²) 等复合函数的导数。识别外层和内层函数，应用链式法则，并清楚保留内层导数。

Free core calculationsAnimated rule sequenceDeterministic verification

专属计算工作台

导数求解器

计算引擎

使用 ^ 表示乘方，函数请使用括号，例如 sin(x)、log(x) 或 exp(2x)。

试试这些例题

Understand the nested structure

链式法则如何工作

A useful chain rule derivative calculator reveals which function is outside, which expression is inside, and why both derivatives are required.

Recognize a composite function

The chain rule applies when one function is evaluated inside another. In (x²+1)³, the cube is outside and x²+1 is inside. In sin(x²), sine is outside and x² is inside.

Rename the inner expression u when the layers feel crowded. Then differentiate the outside with respect to u, restore the original inner expression, and multiply by du/dx.

Differentiate the outside, keep the inside

For (x²+1)³, treat x²+1 as one object. The outside power becomes 3(x²+1)². Multiplying by the inner derivative 2x gives 6x(x²+1)².

For sin(x²), the outer derivative is cos(x²) and the inner derivative is 2x. For exp(3x+1), the exponential remains and the linear inside contributes 3.

Handle multiple nested layers

Expressions such as sin((x²+1)³) contain three layers. Work from outside to inside and retain a derivative factor for every layer instead of skipping directly to a simplified answer.

A chain rule may also sit inside a product or quotient. Apply the top-level rule first, then use the chain rule separately wherever a factor contains a composition.

Check the rule structure

A correct result contains the derivative of the outside evaluated at the unchanged inside, followed by the inner derivative. Missing that final factor is the most common chain-rule error.

CalculusMate recomputes supported first derivatives with its deterministic tree. Equivalent answers can look different, so compare the rule structure before comparing formatting.

A practical chain-rule workflow

Start by reading the expression from the outside inward. Ask which operation would be performed last if you evaluated the function at a number. That operation is the outer layer. In (x²+1)³, you would first square x, add one, and cube the result, so the cube is outside. In sin(x²), sine acts on the completed square. This reading habit is more dependable than searching for a memorized visual pattern. It also scales to roots, logarithms, exponentials, and several nested layers. Next, give the inner expression a temporary name such as u. The substitution is not required algebraically, but it separates two jobs that students often mix together. Differentiate the outer function with respect to u, then multiply by du/dx. For y=(x²+1)³, write y=u³ and u=x²+1. The two derivatives are dy/du=3u² and du/dx=2x. Their product becomes 6x(x²+1)² after u is restored. A chain rule derivative calculator should expose both factors so you can see exactly where each part came from. For a longer composition, repeat the same decision at every layer. If y=sin((x²+1)³), the derivative has a cosine factor from sine, a 3(x²+1)² factor from the cube, and a 2x factor from the quadratic. Working outside to inside creates cos((x²+1)³)·3(x²+1)²·2x. You may simplify the constants afterward, but delaying simplification makes missing layers easier to detect. The same logic explains why d/dx exp(sin x)=exp(sin x)·cos x and why d/dx log(1+x²)=2x/(1+x²). Do not confuse composition with multiplication. The expression sin(x²) is a composition because x² is the input of sine. The expression x²sin(x) is a product because two factors are multiplied. A problem can require both rules: differentiating x²sin(x³) begins with the product rule, while the derivative of sin(x³) inside its second term uses the chain rule. Quotients can contain composite functions in the same way. Identify the top-level structure first, then solve each nested piece with the appropriate rule. A quick verification is to inspect the expected factors before simplifying. The outside derivative must still contain the unchanged inside, and a factor representing the derivative of every nontrivial inner layer must appear. You can also compare numerical slopes near a regular point: evaluate the function slightly to either side and see whether the sign and approximate size agree with the symbolic derivative. Numerical checks do not prove an identity, but they catch many dropped signs and missing inner derivatives. CalculusMate uses deterministic symbolic rules for supported inputs and reports unsupported structures instead of filling a gap with generated prose. When writing a final solution, keep one line for identifying u, one for the chain-rule formula, one for the two derivatives, and one for substitution and simplification. This format is short enough for homework but detailed enough to audit. The animated sequence on this page follows the same order. It is designed to help you understand why dy/dx contains each factor, not merely to provide a result to copy.

Practice without hiding the structure

Try the same inner function under several outer functions: (x²+1)⁴, sin(x²+1), exp(x²+1), and log(x²+1). The inner derivative 2x repeats while the outside factor changes. Then reverse the exercise by keeping sine outside and changing its input. This isolates what each layer contributes. Before opening the answer, predict the number of derivative factors and their signs. After solving, expand only if the question requires it; factored chain-rule forms are often easier to verify and use in later algebra. If two forms look different, substitute a simple regular value or simplify their ratio rather than assuming one is wrong. Finally, explain the solution aloud in the order outside, inside, multiply. Being able to name those three moves is a stronger test of understanding than reproducing one memorized formula.

Worked chain-rule patterns

不只看答案，也看清代换过程

d/dx (x²+1)³

Let u=x²+1. Differentiate u³, restore the inside, and multiply by 2x to obtain 6x(x²+1)².

d/dx sin(x²)

Keep x² inside cosine, then multiply by the derivative of x². The result is 2x·cos(x²).

d/dx exp(3x+1)

The exponential reproduces itself while the inner linear function contributes a factor of 3.

Supported chain-rule boundaries

The engine handles supported powers and common trigonometric, exponential, logarithmic and root compositions. Variable exponents and specialized functions may be reported as unsupported rather than guessed.

FAQ
What is the chain rule?+

For y=f(g(x)), dy/dx=f′(g(x))·g′(x): differentiate the outside at the unchanged inside, then multiply by the inner derivative.

When do you use the chain rule instead of the product rule?+

Use the chain rule for nested functions and the product rule for multiplied factors. Some expressions require both.

Does it show every step for free?+

Supported core calculations show their ordered rule path without sign-up. Small algebraic simplifications may be grouped.

Can it solve nested chain rules?+

Yes, when each nested operation belongs to the current deterministic rule set.