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

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

So, without wasting time let's get started.

## What is cos x?

Cos x is a trigonometric function that is reciprocal of sec x.

## Derivative of cos x

The derivative of cos x is equal to the negative of sin x.

We can prove the derivative of cos 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 cos 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

Let us,

y = cos x

As we know,

cos x  = 1/sec x

So,

We can written as,

y  = 1/sec x.

Where,

u = 1

v = sec x

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

dy/dx = [sec x d/dx(1) - 1 d/dx(sec x)] / (sec x)²

Since,

d/dx (sec x) = sec x .tan x and d/dx (1) = 0

So,

dy/dx = (0 - sec x . tan x )/ (sec x)²,

dy/dx = - tan x/sec x

As we know,

tan x = (sin x/ cos x) and sec x = 1/cos x

So,

dy/dx  =  - (sin x/ cos x)/(1/cos x)

d/dx ( cos x) = - sin x

Thus, we proved the derivative of cos x will be equal to - sin x using the quotient rule method.

### Derivative of cos 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) = cos x

So,

f(x + h) = cos (x + h)

Putting these values on the above first principle rules equation.

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

So, as we know

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

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

= limₕ→₀[ {(cos h - 1)/h}cos x - (sin h/h)sin x]

= limₕ→₀( 0.cos x - 1 . sin x)

f' (x)  = - sin x

Thus, we proved the derivative of cos x will be equal to - sin x using the first principle rule method.

### Derivative of cos x Proof by Chain Rule

Let us,

y = cos x

As we know,

sin{(π/2) - x} = cos x

So,

y = sin{(π/2) - x}

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 = d/dx [sin{(π/2) - x}]

Since, d/dx(sin x) = cos x

dy/dx =  [cos{(π/2) - x}] . (-1)

Since, cos{(π/2) - x} =  sin x

Hence,

d/dx ( cos x) = - sin x

Thus, we proved the derivative of cos x will be equal to - sin x using the chain rule method.

## FAQ Related to the Derivative of cos x

### How to Find the Derivative of cos x?

The derivative of cos x can be derived by different methods such as quotient rule, first principle rule, chain rule.

### What is the Second Derivative of cos x?

To find the second derivative of cos x, we have to find the first derivative of cos x, which is -sin x.  Then find the derivative of -sin x which is -cos x.  Therefore, the second derivative of cos x will be -cos x.

### What is the derivative of cos x?

The derivative of cos x is equal to the negative of sin x.

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

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