differentiation of e^(x^2)?

Question

what is the approach through first principles?

Answer 1

The formula to be used is d/dx e^u = e^u d/dx(u).
Thus, if
y = e^(x^2)
y' = e^(x^2) d/dx(x^2)
= e^(x^2)*2x
= 2xe^(x^2)

Answer 2

let u=x^2
the derivative of e^x is e^x
y=e^u
dy/dx=e^u(du/dx)
=e^u times (2x)
=2xe^(x^2)

Answer 3

d e ^ f(x) / dx = f '(x) * e^f(x)

'differentiate by parts'