# Derivative of tan x: Proof by Quotient, Chain & First Principle

In this article, you will learn what is the derivative of tan x as well as prove the derivative of tan x by chain rule, first principal rule, and quotient rule.

I have already discussed the derivative of sin x, cos x, and sec x in a previous article you can check from here.

So, without wasting time let's get started.

## What is tan x?

tan x is a trigonometric function which is equal to the (sin x)/(cos x).

This trigonometric function is mostly used in right-angle triangles.

## Derivative of tan x

The derivative of tan x is equal to the sec²x.

We can prove the derivative of tan x in three ways first by using the quotient rule and second by using the first principle rule and the last chain rule.

### Derivative of tan x Proof by Quotient Rule

The formula of the quotient rule is,

dy/dx =  {v (du/dx) - u (dv/dx)}/v²

Where,

dy/dx = derivative of y with respect to x

v = variable v

du/dx = derivative of u with respect to x

u = variable u

dv/dx = derivative of v with respect to x

v = variable v

As we know,

tan x = sin x/cos x

So,

Let us

y = tan x

and,

u = sin x

v = cos x

Now putting these values on the quotient rule formula, we will get

dy/dx = [ cos x × d/dx (sin x) - sin x × d/dx (cos x)] / (cos²x)

= [cos x · cos x - sin x (-sin x)] / (cos²x)

= [cos²x + sin²x] / (cos²x)

So, from the Pythagoras theorem, we know that cos²x + sin²x = 1

So,

dy/dx = 1 / cos²x

So, 1 / cos²x = sec²x

Hence,

d/dx(tan x) = sec²x

Thus, we proved the derivative of tan x will be equal to sec²x using the quotient rule method.

### Derivative of tan x Proof by First Principle Rule

According to the first principle rule, the derivative limit of a function can be determined by computing the formula:

For a differentiable function y = f (x)

We define its derivative w.r.t  x as :

dy/dx = f ' (x) = limₕ→₀ [f(x+h) - f(x)]/h

f'(x) = limₕ→₀ [f(x+h) - f(x)]/h

This limit is used to represent the instantaneous rate of change of the function f(x).

Let,

f (x) = tan x

So,

f(x + h) = tan (x + h)

Putting these values on the above first principle rules equation.

f'(x) = limₕ→₀ [tan(x + h) - tan x] / h

Since, we know tan x = sin x/cos x

So, putting these values

f'(x) = limₕ→₀ [ [sin (x + h) / cos (x + h)] - [sin x / cos x] ] / h

= limₕ→₀ [ [sin (x + h ) cos x - cos (x + h) sin x] / [cos x · cos(x + h)] ]/ h

As we know,

sin (a - b) = sin a cos b - cos a sin b

So,

f'(x) = limₕ→₀ [ sin (x + h - x) ] / [ h cos x × cos(x + h)]

= limₕ→₀ [ sin h ] / [ h cos x × cos(x + h)]

= limₕ→₀ (sin h)/ h limₕ→₀ 1 / [cos x × cos(x + h)]

So, after applying the limit,

limₕ→₀ (sin h)/ h = 1.

f'(x)  = 1 [ 1 / (cos x × cos(x + 0))]

f'(x)  = 1/cos² x

As we know,

1/cos x = sec x

So,

f'(x) = sec²x.

Thus, we proved the derivative of tan x will be equal to sec²x using the first principle rule method.

### Derivative of tan x Proof by Chain Rule

Let us,

y = tan x

As we know,

tan x = 1/cot x

So,

y = 1 / (cot x)

= (cot x)⁻¹

By using the chain rule,

The formula of chain rule is,

dy/dx = (dy/du) × (du/dx)

Where,

dy/dx = derivative of y with respect to x

dy/du = derivative of y with respect to u

du/dx = derivative of u with respect to x

After putting these values we can find,

dy/dx  =  -1 (cot x)⁻² × d/dx (cot x)

Since,

d/dx (cot x) = (-cosec²x) and a⁻ⁿ = 1/aⁿ

So,

dy/dx = -1/(cot² x) × (-cosec²x)

As we know,

1/(cot² x) = tan² x

So,

dy/dx =  (tan² x) × (cosec²x)

Now we know that the trigonometric formula

tan x = sin x/cos x  and cosec x = 1/sin x

So after putting these values we will get,

dy/dx = (sin² x)/(cos² x) × (1/sin² x)

= 1/cos² x

As we know,

1/cos x = sec x

d/dx (tan x)  = sec²x

Thus, we proved the derivative of tan x will be equal to sec²x using the chain rule method.

## FAQ Related to Derivative of tan x

### What is a derivative of tan x?

The derivative of tan x is equal to the sec²x.

### Where is tan equal to 1?

At value tan 45° is equal to 1.

### What is tan Infinity?

At value tan 90° is equal to infinity.

### What is the differentiation of tan x?

The differentiation of tan x is equal to the sec²x.

So friends here I discussed all aspects related to the derivative of tan x

I hope you enjoy this topic If you have any doubt then you can ask me through comments or direct mail.

Thank You.