How to apply the chain rule when the inner function is a product?

[JonathanW]

I keep getting tangled up when I try to apply the chain rule to functions where the inner function itself is a product, like f(g(x)*h(x)). My textbook just shows the standard form, but when I differentiate something like sin(x² * e^x), I'm not sure if I'm handling the derivative of the inner product correctly before multiplying by the outer derivative.

[Ryan_J]

You're right to check the inner derivative first. For sin(x^2 e^x), set u = x^2 e^x. Then du/dx = e^x(2x) + x^2 e^x = e^x(2x + x^2). By the chain rule, d/dx sin(u) = cos(u) * du/dx. So the derivative is cos(x^2 e^x) * e^x(2x + x^2).

[Aubrey.J]

I tried it once and got tangled too. I did d/dx of x^2 e^x as 2x e^x + x^2 e^x, then multiplied by cos(x^2 e^x). It felt right, but I kept wondering if I should distribute differently or miss a term.

[Aaron.G]

Is the real snag that the inner x^2 e^x is itself a product, so you treat it with its own product rule first and only then use the outer derivative?

[NicholasT]

I remember staring at sin((x^2)(e^x)) and drifting into a tangent about the graph, then coming back and realizing the inner derivative has two terms and you just multiply by the outer cos.