Question

what is the derivative of 5cos^2(pi)(t)?

Answer 1

i assume you mean 5cos^2((pi*t)) in which case;
i would use the chain rule,
subs u=cos(pi*t), so you have (i presume it's y), y=5u^2
dy/du = 10u and du/dt=-pi*sin(pi*t)
Multiply them together,
dy/dx = 10u * -pi*sin(pi*t)
replace u=cos(pi*t)
dy/dx=-10*pi*cos(pi*t)*sin(pi*t)
And you could then use the relation that sin(2x)=2sin(x)cos(x)
so
dy/dx=-5*pi*sin(2*pi*t)

You could alternatively use the relation that 2cos(2x)=cos^2(x)+1 and rearrange for cos^2(x) [x being pi*t] .. which is a better way if you know how to do it .. hope this helps :)