Knowing that for every function h(x) that can be written as f(g(x)), h^(')(x) = f^(')(g(x))*g^(')(x)
In this case we have f(x) = tan(x) and g(x) = (pi*x)/2.
The derivative of g(x) is easily computed, since the derivative of a first degree polynomial is the leading coefficient, so:
g^(')(x) = pi/2
The derivative of f(x) you can either check from a reference table (or remember it) since it shows up reasonably a lot, or use the quotient rule:
tan(x) = sin(x)/cos(x) rarr tan^(')(x) = (cos^2(x)+sin^2(x))/cos^2(x)
Using the identity cos^2(x) + sin^2(x) = 1, we get that:
tan^(')(x) = 1/cos^2(x) = sec^2(x)
So the derivative of tan((pi*x)/2) = pi/2*sec^2((pi*x)/2)
