How do I apply the chain rule to sin(x^2) correctly?

[OliviaUS]

I’m really struggling to understand how to apply the chain rule when the function is something like sin(x²). I get the basic idea of outside and inside derivatives, but when I try to work through a problem, my steps get muddled and I’m not sure if I’m multiplying the derivatives in the correct order.

[EvelynT]

I wrestled with the chain rule for sin x squared last week too. I set u equal to x squared so y equals sin u. Then dy over du equals cos u and du over dx equals 2x. Multiply them and you get dy over dx equals 2x cos x squared. I checked at x equals 0.5 by plugging into a calculator and the slope matched.

[ScottEG]

I still get tangled when I try to do it in my head. Sometimes I accidentally differentiate sin instead of grabbing the inside function. I try to keep it by writing the inside as a separate function first and then differentiate both pieces.

[JonathanW]

I did a quick numeric check by a tiny step test I used h small and I compared sin x plus h all squared minus sin x all squared divided by h with 2x cos x squared It lines up and the main win was writing it as inner function x squared and seeing the two parts clearly

[AbigailB]

Do you think the real issue is that you are solving the right problem or maybe the mismatch is somewhere else in the algebra?