Proof of Derivative of Sin x

The derivative of sin x can be proved using the following ways:

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

Derivative of Sin x by First Principle of Derivative

To prove derivative of sin x using First Principle of Derivative, we will use basic limits and trigonometric formulas which are listed below:

1. sin (x + y) = sin x cos y + sin y cos x
2. lim _x→0 [sin x/x] = 1
3. lim _x→0 [(cos x – 1)/x] = 0

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

By the First Principle of Derivative

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

⇒ (d/dx) sin x = lim_h→0 [sinx cosh + sinh cosx – sinx]/ h [By 1]

⇒ (d/dx) sin x = lim_h→0 [{sinx (cosh – 1)}/h + {(sinh/h) cosx}]

⇒ (d/dx) sin x = lim_h→0 {sinx (cosh – 1)}/h + lim_h→0{(sinh/h) cosx} [By 2 and 3]

⇒ (d/dx) sin x = sinx (0)+ (1)cosx

⇒ (d/dx) sin x = cosx

Derivative of Sin x by Quotient Rule

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

1. sin x = 1/cosec x
2. (d/dx) [u/v] = [u’v – uv’]/v²

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

y = sin x

y = 1/cosec x

⇒ y’ = (d/dx) [1/cosec x]

Applying quotient rule

y’ = [(d/dx) (1) cosec x – 1.(d/dx)(cosec x)]/(cosec x)²

⇒ y’ = [(0) cosec x – (1) (-cosec x cot x)]/(cosec x)²

⇒ y’ = (cosec x cot x)/(cosec x)²

⇒ y’ = cot x/cosec x

⇒ y’ = (cos x/sin x )/( 1/sin x)

⇒ y’ = cos x

Derivative of Sin 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:

1. sin x = cos [(π/2) – x]
2. cos x = sin [(π/2) – x]

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

y = sin x

y = cos [(π/2) – x] {From Formula 1}

⇒ y’ = (d/dx){cos [(π/2) – x]}

By applying chain rule

y’ = (d/dx){cos [(π/2) – x]}(d/dx)[(π/2) – x]

⇒ y’ = {-sin [(π/2) – x]}(0 – 1)

⇒ y’ = {-sin [(π/2) – x]}(- 1)

⇒ y’ = sin [(π/2) – x]

⇒ y’ = cos x

Also, Check

• Differentiation of Trigonometric Functions
• Derivative of Inverse Trigonometric Functions
• Differentiation Formulas

Derivative of Sin x is Cos x. It refers to the process of finding the change in the sine function with respect to the independent variable. This process is known as differentiation, which is one of the fundamental tools in calculus used to determine the rate of change for various functions. The specific process of finding the derivative for trigonometric functions is referred to as trigonometric differentiation, and the derivative of Sin x is one of the key results in trigonometric differentiation.

In this article, we will learn about the derivative of sin x and its formula including the proof of the formula using the first principle of derivatives, quotient rule, and chain rule as well. Other than that, we have also provided some solved examples for better understanding and answered some FAQs on derivatives of sin x as well. Let’s start our learning on the topic Derivative of Sin x.