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Choosing a derivative rule

Chain Rule vs. Product Rule: How to Tell the Difference

The two rules are often confused because both expressions can look crowded. The reliable test is structural: ask whether one function receives another as its input or whether two changing factors are multiplied.

Direct answer

Use the chain rule for f(g(x)), the product rule for f(x)g(x), and apply both when a multiplied factor contains a composition.

Identify the top-level operation before differentiating: function application means composition, while an explicit product joins separate factors.

Decision method: inspect the top-level operation

In sin(x^2), x^2 sits inside sine. This is composition, so differentiate sine and multiply by the derivative of x^2.

In x^2 sin(x), the factors x^2 and sin(x) are multiplied. This is a product, so each factor is differentiated once while the other is preserved.

In x^2 sin(x^3), start with the product rule. The derivative of the second factor then requires the chain rule because x^3 is inside sine.

Parentheses alone do not determine the rule. Read what operation combines the largest pieces of the expression.

1
Nested: f(g(x)) -> chain rule
2
Multiplied: f(x)g(x) -> product rule
3
Product with a nested factor -> use both

Worked examples

ddx[sin⁡(x2)]\frac{\mathrm{d}}{\mathrm{d}x}\left[\sin\left(x^{2}\right)\right]
2x cos(x^2)

One function is inside another, so use the chain rule.

ddx[x2 sin⁡x]\frac{\mathrm{d}}{\mathrm{d}x}\left[x^{2}\,\sin x\right]
2x sin(x)+x^2 cos(x)

Two changing factors are multiplied, so use the product rule.

ddx[x2 sin⁡(x3)]\frac{\mathrm{d}}{\mathrm{d}x}\left[x^{2}\,\sin\left(x^{3}\right)\right]
2x sin(x^3)+3x^4 cos(x^3)

The top level is a product; the sine factor also needs the chain rule.

Common mistakes

  • • Using the product rule on sin(x^2) because two expressions are visible.
  • • Using only the chain rule on x^2 sin(x) and ignoring the product.
  • • Applying the product rule correctly but forgetting the inner derivative of sin(x^3).
  • • Expanding an expression before identifying its top-level structure.

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

合成関数の微分計算:sin⁡(x2)\sin\left(x^{2}\right)合成関数の微分計算:x2 sin⁡(x3)x^{2}\,\sin\left(x^{3}\right)
Open the calculator without a preset: 合成関数の微分計算

References

Frequently asked questions

Can one problem need both rules?+

Yes. Apply the product rule at the top level, then use the chain rule inside any composite factor.

Are parentheses always a sign of the chain rule?+

No. Parentheses can group a sum or product. The chain rule requires one function to be evaluated at another function.

Which rule should I apply first?+

Apply the rule that matches the top-level operation, then solve nested pieces with their own rules.

Conclusion

Choose the rule from structure, not visual complexity. Nested functions use the chain rule, multiplied factors use the product rule, and some expressions require both in that order.

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