$\frac{d}{ds}$ f(s)

or

f'(s)

This given function is also known as derivative or we can say differentiation with respect to‘s’. In some cases the differentiation of a function f (s) is known as differentiation coefficient of f (s).

Now we will talk about the derivative of tanx.

The derivative of tan x is shown below:

$\frac{d}{dx}$ Tan x

= sec

^{2}x.

Now we will see the proof of derivative of tanx. If we want to find the derivative of tan x than it is necessary to find the derivative of sin x and cos x because we know that tan x = $\frac{sin x}{cos x}$; so by using quotient rule:

So we can write it as:

$\frac{d}{dx}$ Tan x = $\frac{ cos x \frac{d}{dx} sin x - sin x \frac{d}{dx} cos x}{(cos x)^{2}}$

We know that differentiation of sin (x) is cos (x) and differentiation of cos (x) is – sin (x). So put the value of sin (x) and cos (x) in the above expression.

[(cos (x) cos (x) + sin (x) sin (x))] / cos

^{2}(x);

We can write it as:

= 1 + tan

^{2}(x);

We know that 1 + tan

^{2}(x) = sec

^{2}x;

This is all about derivative of tan x.