Find the first differential with respect to x of y=tan(x)

To answer, we must be familiar with several trigonometric identities and expressions; first notice that tan(x)=sin(x)/cos(x). Now our function is a quotient of two functions of x that we can easily differentiate. Using the quotient rule gives dy/dx=[cos(x)cos(x)-sin(x)(-sin(x))]/cos^2(x). The numerator simplifies into cos^2(x)+sin^2(x), which our trigonometric identities tell us is just equal to 1. Hence we have dy/dx=1/cos^2(x), and as sec(x)=1/cos(x), we can express this as dy/dx=sec^2(x).