We'll have to use the chain rule here. The chain rule, in plain English, says that the derivative of a compound function (like tan(x^2), (which is the x^2 function inside the tan(x) function - that's what makes it compound), is the derivative of the "inside" function multiplied by the derivative of the entire function. In mathspeak, we say the derivative of f(g(x))=f'(g(x))*g'(x).
In our case, the "inside" function is x^2, and the derivative of x^2 is, of course, 2x. The entire function is tan(x^2), and we know the derivative of tan(x) is sec^2(x); so the derivative of tan(x^2) is sec^2(x^2). Multiplying these two derivatives together gives 2xsec^2(x^2), which is our final result.
