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.

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

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

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.

See Also:



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

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

Thank You.

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