**In trigonometry, we will study about different types of functions. Here, we will see the different types of derivatives of trigonometric functions**.

Let us see the different types of derivatives of inverse in the trigonometric functions:

$\frac{d}{du}$ sin (u) = cos (u)

$\frac{d}{du}$ cos (u) = - sin (u)

$\frac{d}{du}$ tan (u) = sec

^{2}(u)

$\frac{d}{du}$ csc (u) = - csc (u) cot (u)

$\frac{d}{du}$ sec (u) = sec (u) tan (u)

$\frac{d}{du}$ cot (u) = - csc

^{2}(u)

Now, we will see the derivative of sec x tan x and see how to prove the derivative of sec x tan x.

The derivative of Sec x Tan x is sec (x) (tan

^{2}(x) + sec

^{2}(x)). Here, we will see the proof of the derivative of Sec x Tan x.

First, write the given function in the derivative form as follows:

$\frac{d}{du}$ sec x tan x = sec (x) (tan

^{2}(x) + sec

^{2}(x))

Here, we have to follow the product rule for finding the derivative of the given function.

The product rule is given by:

(u v)' = uv' + u'v

Here, the value of u is sec (x) and the value of v = tan (x).

Put these values in the product rule. So, we can write it as follows:

$\frac{d}{dx}$ sec x tan x = sec x $\frac{d}{dx}$ tan x + tan x $\frac{d}{dx}$ sec x

If we differentiate tan x with respect to x, we get sec

^{2}x and the differentiation of sec x is sec x tan x.

So, we can write it as follows:

sec x

^{3}+ tan (x) (tan (x) sec (x))

Or, we can write it as follows:

sec (x) (tan

^{2}(x) + sec

^{2}(x))

So, the derivative of sec x tan x is sec (x) (tan

^{2}(x) + sec

^{2}(x))