How do I differentiate tan(x) ?

To differentiate tan(x):

Note: Here, we use d/dx f(x) to mean "the derivative of f(x) with respect to x". 

1) rewrite tan(x) as sin(x)/cos(x)

2) Apply the quotient rule (or, alternatively, you could use the product rule using functions sin(x) and 1/cos(x)):

Using the quotient rule:

d/dx tan(x) = (cos(x)cos(x) - sin(x)(-sin(x))) / cos²(x)

d/dx tan(x) = (cos²(x) + sin²(x)) / cos²(x)

3) Recall/Note the following identity: cos²(x) + sin²(x) = 1

So, d/dx tan(x) = 1 / cos²(x)

4) Use the definition of sec(x):

So, d/dx tan(x) = sec²(x), as required