Question

What is the derivative of `y=tan(x)` ?

Answer 1

The derivative of `tanx` is `sec^2x`.

To see why, you'll need to know a few results. First, you need to know that the derivative of `sinx` is `cosx`. Here's a proof of that result from first principles:

Once you know this, it also implies that the derivative of `cosx` is `-sinx` (which you'll also need later). You need to know one more thing, which is the Quotient Rule for differentiation:

Once all those pieces are in place, the differentiation goes as follows:

`d/dx tanx`
`=d/dx sinx/cosx`

`=(cosx . cosx-sinx.(-sinx))/(cos^2x)` (using Quotient Rule)

`=(cos^2x+sin^2x)/(cos^2x)`

`=1/(cos^2x)` (using the Pythagorean Identity)

`=sec^2x`