I keep mixing these up on derivative questions. Is there a quick way to decide before I start differentiating?
Accepted answer
Look at how the expression is *built*.
- **Product rule** — two functions multiplied side by side: `f(x)·g(x)`, like `x²sin(x)`. Derivative: `f'g + fg'`.
- **Chain rule** — one function *inside* another: `f(g(x))`, like `sin(x²)`. Derivative: `f'(g(x))·g'(x)`.
Ask yourself: could I evaluate this by computing two separate things and multiplying (product), or do I have to finish an inner step before the outer one can act (chain)? `x²sin(x)` is a product; `sin(x²)` requires squaring before the sine, so it's a chain. Nested products like `x²sin(x³)` need both.