Proof of Derivative of Cos x

The derivative of cos x can be derived using the following ways:

• By using the First Principle of Derivative
• By using Chain Rule
• By using Quotient Rule

Derivative of Cos x by First Principle of Derivative

Let us study the derivation of cos x using the First Principle of derivative i.e., the definition of limits. Here, x approaches x + h and the limiting value approaches 0.

To prove it we must know some basic trigonometric formulas:

• cos (A + B) = cos A cos B – sin A sin B
• lim _x→0 [(cos x – 1) / x] = 1
• lim _x→0 [sin x/x] = 1

Now, let’s see the proof of it:

(d/dx) cos x = lim_h→0 [cos(x + h) – cos x]/[(x + h) – x]

⇒ (d/dx) sin x = lim_h→0 [cos x cos h – sin x sin x – cos x]/ h

⇒ (d/dx) sin x = lim_h→0 [{(cos h – 1) / h} cos x – {(sin h/h) sin x}]

⇒ (d/dx) sin x = cos x (0) – (1) sin x

⇒ (d/dx) sin x = -sin x

Derivative of Cos x by Chain Rule

To prove derivative of sin x using chain rule, we will use basic derivatives and trigonometric formulas which are listed below:

• sin x = cos [(π/2) – x]
• cos x = sin [(π/2) – x]
• d(sin x)/dx = cos x

Let’s see the proof of it by chain rule:

By applying chain rule, we have

y’ = (d/dx){sin [(π/2) – x]}

⇒ y’ = {cos [(π/2) – x]} (– 1)

⇒ y’ = -cos [(π/2) – x]

⇒ y’ = -sin x

Derivative of Sin x by Quotient Rule

The basic formula you must know before proving derivative of Sin x by Quotient Rule are:

• cos x = 1/sec x
• sec x = 1/cos x
• d(sec x)/dx = sec x tan x
• (d/dx) [u/v] = [u’v – uv’]/v²
• tan x = sin x/ cos x

Let’s start the proof of the derivative of sin x:

y’ = (d/dx) (1/sec x)

⇒ y’ = [(1)’ sec x – 1.(sec x)’]/(sec²x)

⇒ y’ = [(0) sec x – (sec x tan x)]/(sec² x)

⇒ y’ = (-sec x tan x)/(sec² x)

⇒ y’ = -tan x/sec x

⇒ y’ = (-sin x/cos x )/( 1/cos x)

⇒ y’ = -sin x

Derivative of Cosine Function, cos(x), with respect to x is -sin x. Derivative of Cos x is the change in the cosine function with respect to the variable x and represents its slope at any point x. Thus, in other words, we can say that the slope of cos x is – sin x for all real values x.

In this article, we will learn about the derivative of Cos x and its formula including the proof of the formula using the first principle of derivatives, quotient rule, and chain rule. Solved Problems and FAQs are also provided in the end along with some practice questions to learn the topic more clearly.