Derivative of Sin x
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.
Table of Content
- Derivative in Math
- What is Derivative of Sin x?
- Proof of Derivative of Sin x
- Solved Examples
- Practice Questions
What is Derivative in Math?
The derivative of a function is the rate of change of the function with respect to any independent variable. The derivative of a function f(x) is denoted as f'(x) or (d /dx)[f(x)]. The differentiation of a trigonometric function is called as derivative of the trigonometric function or trig derivatives.
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What is Derivative of Sin x?
Among the trig derivatives, the derivative of the sinx is one of the derivatives. The derivative of the sin x is cos x. The derivative of sin x is the rate of change with respect to angle i.e., x. The resultant of the derivative of sin x is cos x.
Derivative of Sin x Formula
The formula for the derivative of sin x is given by:
(d/dx) [sin x] = cos x
or
(sin x)’ = cos x
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:
- sin (x + y) = sin x cos y + sin y cos x
- lim x→0 [sin x/x] = 1
- 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 = limh→0 [sin(x + h) – sinx]/[(x + h) – x]
⇒ (d/dx) sin x = limh→0 [sinx cosh + sinh cosx – sinx]/ h [By 1]
⇒ (d/dx) sin x = limh→0 [{sinx (cosh – 1)}/h + {(sinh/h) cosx}]
⇒ (d/dx) sin x = limh→0 {sinx (cosh – 1)}/h + limh→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:
- sin x = 1/cosec x
- (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:
- sin x = cos [(π/2) – x]
- 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
Solved Examples on Derivative of Sin x
Example 1: Find the derivative of sin 4x.
Solution:
Let y = sin 4x
⇒ y’ = (d/dx) [sin 4x]
Applying chain rule
y’ = (d/dx) [sin 4x].(d/dx) (4x)
⇒ y’ = (cos 4x)4
⇒ y’ = 4cos4x
Example 2: Evaluate the derivative f(x) = (x3 + 5x2 + 2x + 7)sinx
Solution:
f(x) = (x3 + 5x2 + 2x + 7)sinx
⇒ f'(x) = (d /dx)[(x3 + 5x2 + 2x + 7)sinx]
Applying product rule
f'(x) = (d /dx)[(x3 + 5x2 + 2x + 7)]sinx + (x3 + 5x2 + 2x + 7)(d /dx)[sinx]
⇒ f'(x) = (3x2 + 10x +2)sinx + (x3 + 5x2 + 2x + 7)cosx
Example 3: Find the derivative of p(x) = (4x2 + 9)/sinx
Solution:
p(x) = (4x2 + 9)/sinx
⇒ p'(x) = (d /dx)[(4x2 + 9)/sinx]
Applying quotient rule
p'(x) = [(d /dx)(4x2 + 9) sin x – (4x2 + 9)(d /dx)sin x]/ sin2x
⇒ p'(x) = [8xsin x – (4x2 + 9)cos x]/ sin2x
Example 4: Find the derivative of the function (cosx)sinx
Solution:
Let y = (cosx)sinx
Taking log
ln y = ln (cosx)sinx
⇒ ln y = (sin x) ln (cos x)
Differentiating the above equation, we get
(1/y) y’ = (d/dx)[(sin x) ln (cos x)]
Applying product rule
(1/y) y’ = (d/dx)(sin x) [ln (cos x)]+ (sin x)(d/dx)[ln (cos x)]
⇒ (1/y) y’ = cos x [ln (cos x)]+ (sin x)[(-sinx)/(cos x)]
⇒ (1/y) y’ = cos x {ln (cos x)} – sin x tan x
⇒ y’ = y[cos x {ln (cos x)} – sin x tan x]
⇒ y’ = (cosx)sinx [cos x {ln (cos x)} – sin x tan x]
Example 5: Evaluate the derivative sin 5x + x.sinx
Solution:
Let z = sin 5x + xsinx
Differentiating
z’ = (d/dx) [sin 5x + xsinx]
⇒ z’ = (d/dx) sin 5x + (d/dx)[xsinx]
Applying chain rule and product rule
z’ = 5 cos 5x + (d/dx)(x)sinx + x(d/dx)(sinx)
⇒ z’ = 5 cos 5x + sinx + xcosx
Example 6: Find derivative of sin-1 x.
Solution:
(d /dx) [sin-1 x] = 1/√[1 – x2] [From Formula]
Example 7: Find derivative of sin x2
Solution:
By applying chain rule
(d/dx) [sin x2] = (d/dx) [sin x2](d/dx) [x2]
⇒ (d/dx) [sin x2] = [cos x2][2x]
⇒ (d/dx) [sin x2] = 2x cos x2
Example 8: Find derivative of sin x. cos x
Solution:
By applying product rule
(d/dx) [sin x. cos x] = (d/dx) [sin x] cos x + sin x (d/dx) [cos x]
⇒ (d/dx) [sin x. cos x] = cos x. cos x + sin x (-sin x)
⇒ (d/dx) [sin x. cos x] = cos2 x – sin2 x
⇒ (d/dx) [sin x. cos x] = cos 2x
Example 9: Find derivative of x sin x
Solution:
By applying product rule
⇒ (d/dx) [x sin x] = (d/dx) [x] sin x + x (d/dx) [sin x]
⇒ (d/dx) [x sin x] = (1) sin x + x cos x
⇒ (d/dx) [x sin x] = sin x + x cos x
Example 10: What is derivative of sin x and cos x
Solution:
(d/dx) sin x = cos x
⇒ (d/dx) cos x = -sin x
Practice Questions on Derivative of Sin x
Q1. Find the derivative of sin 7x
Q2. Find the derivative of x2.sinx
Q3. Evaluate: (d/dx) [sin x/(x2 + 2)]
Q4. Evaluate the derivative of: sin x. tan x
Q5. Find: (tan x)sin x
Derivative of Sin x FAQs
What is Derivative?
The derivative of the function is defined as the rate of change of the function with respect to a variable.
Write the Formula for Derivative of Sin x.
The formula for derivative of sin x is:
(d/dx) sin x = cos x
What is the Derivative of Sin (-x)?
Derivative of sin (-x) is -cos (-x) or -cos x.
What are the Different Methods to Prove Derivative of Sin x?
The different methods to prove derivative of sin x are:
- By using First Principle of Derivative
- By Quotient Rule
- By Chain Rule
What is the Derivative of Negative Sin x?
Derivative of negative sin x i.e., -sin x; is -cos x.
What is Derivative of Cos x?
Derivative of cos x -sin x.
What is the Derivative of 2 sin x?
Derivative of 2 sin x is 2 cos x.
What is the Derivative of Tan x?
Derivative of tan x is sec2 x.
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