is there an equivalence rule for sin(x^2)?

Question

I'm helping my brother with his math hw and I think there is some rule for what sin(x^2) is but I can't remember. Either that or as it relates to differentiating sin(x^2). Do we even care that x is being squared when inside the sin function as opposed to sin^2(x)?

Answer 1

Yes, you do care, since sin² x = (sin x)² ≠ sin (x²). These are not the same function.

The way you differentiate composed functions such as sin (x²) is by using the chain rule

d(sin (x²))/dx
cos (x²) · d(x²)/dx
cos (x²) · 2x
2x cos (x²)

Answer 2

yes it is different.

sin(x^2)
let's substitute x=5
sin(25) = .42


now do sin^2(x)
which is [ sin(5) ]^2 = .0076

see they're different.


what are you doing with it? if you are differentiating use u=x^2

then to differentiate it's sin(u)u'