Derivative of Arcsin x – Formula & Proof using Quotient Rule, Chain Rule, and First Principle
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The derivative of arcsin x is 1/√1-x². It is written as d/dx(arcsin x) = 1/√1-x². Arcsin function is the inverse of the sine function and is a pure trigonometric function. We will learn how to differentiate arcsin x by using various differentiation rules like the first principle of derivative, differentiate arcsin x using chain rule and differentiate arcsin x using the quotient rule. Arcsin of x is defined as the inverse sine function of x when -1 ≤ x ≤ 1.
What is Derivative of arcsin x?
Derivative of arcsin function is denoted by d/dx(arcsin x) and its value is 1/√1-x². It returns the angle whose sine is a given number.
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When the sine of y is equal to x:
sin y = x
Then the arcsine of x is equal to the inverse sine function of x, which is equal to y:
Example:
Graph of Arcsine: Arcsin x can be represented in graphical form as follows:
Values of Arcsin
x |
arcsin(x) (rad) |
arcsin(x) (°) |
-1 |
-π/2 |
-90° |
|
-π/3 |
-60° |
|
-π/4 |
-45° |
|
-π/6 |
-30° |
0 |
0 |
0° |
|
π/6 |
30° |
|
π/4 |
45° |
|
π/3 |
60° |
1 |
π/2 |
90° |
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Proof of Derivative of Arcsin x
We will learn how to differentiate arcsin x by using various differentiation rules:
- Proof of Derivative of Arcsin by Quotient Rule
- Proof of Derivative of Arcsin by first principle of derivative
- Proof of Derivative of Arcsin by differentiating arcsin x using chain rule
Proof of Derivative of Arcsin by Quotient Rule
We can prove the derivative of arcsin by quotient rule using the following steps:
Step 1: Write sin y = x,
Step 2: Differentiate both sides of this equation with respect to x.
\(\begin{matrix}
{d\over{dx}}sin y = {d\over{dx}}x\\
cosy {d\over{dx}} y = 1
\end{matrix}\)
Step 3: Solve for
Step 4: Define cos y in terms of x using a reference triangle.
From the reference triangle, the adjacent side is
Step 5: Substitute for cos y.
Step 6: Define arcsine.
Now we can define arcsine as:
Step 7: Differentiate and write the result.
Proof of Derivative of Arcsin by Chain Rule
We can prove the derivative of arcsin by Chain rule using the following steps:
\(\begin{matrix}
\text{ Let } y = arcsin x\\
\text{ Taking sin on both sides, }\\
sin y = sin (arcsin x)\\
\text{ By the definition of an inverse function, we have, }\\
sin (arcsin x) = x\\
\text{ So the equation becomes }
sin y = x \\
\text{ Differentiating both sides with respect to x,}\\
{d\over{dx}} (sin y) = {d\over{dx}} (x)\\
cosy {d\over{dx}} y = 1\\
{d\over{dx}} y = {1\over{cosy}}\\
\text{ Using one of the trigonometric identities }\\
sin^y + cos^y = 1\\
{\therefore} cos y = \sqrt{1 – sin^2y} = \sqrt{1 – x^2}\\
{dy\over{dx}} = {1\over{\sqrt{(1 – x^2)}}}\\
\text{ Substituting y = arcsin x back }\\
{d\over{dx}}(arcsin x) = arcsin’x = {1\over{\sqrt{1-x^2}}}
\end{matrix}\)
Proof of Derivative of Arcsin by First Principle
We can prove the derivative of arcsin by First Principle using the following steps:
\(\begin{matrix}
f’(x)={dy\over{dx}}=\lim _{h{\rightarrow}0}{f(x+h)–f(x)\over{h}}
f(x)=arcsin x\\
f(x+h)=arcsin(x+h)\\
f(x+h)–f(x)= tan(x+h) – tan(x) = arcsin (x + h) – arcsin x\\
{f(x+h) – f(x)\over{h}}={ arcsin (x + h) – arcsin x\over{h}}\\
\lim _{h{\rightarrow}0}{f(x+h) –f(x)\over{h}} = \lim _{h{\rightarrow}0} {arcsin (x + h) – arcsin x\over{h}}\\
\text{ Assume that arcsin (x + h) = A and arcsin x = B }\\
sin A = x + h\\
sin B = x\\
sin A – sin B = (x + h) – x\\
sin A – sin B = h\\
If \text{ h → 0, (sin A – sin B) → 0 sin A → sin B or A → B or A – B → 0}\\
\lim _{A-B{\rightarrow}0}{f(x+h) –f(x)\over{h}} = \lim _{A-B{\rightarrow}0} {(A – B)\over{(sin A – sin B)}}\\
\text{ sin A – sin B = 2 sin [(A – B)/2] cos [(A + B)/2] }\\
\lim _{A-B{\rightarrow}0}{f(x+h) –f(x)\over{h}} = \lim _{A-B{\rightarrow}0} {(A – B)\over{[2 sin [(A – B)/2] cos [(A + B)/2]]}}\\
\text{ A – B → 0, we can have (A – B)/2 → 0 }\\
\lim _{{A-B\over{2}}{\rightarrow}0}{f(x+h) –f(x)\over{h}} = \lim _{{A-B\over{2}}{\rightarrow}0}{1\over{(sin [(A – B)/2]\over{[(A – B)/2])}}} \lim _{A-B{\rightarrow}0} cos[(A + B)/2]\\
f’(x) = cos[(B + B)/2] = cos B\\
sin B = x\\
cos B = \sqrt{1 – sin^2B} = \sqrt{1 – x²}\\
\lim _{h{\rightarrow}0}{f(x+h) –f(x)\over{h}} = {1\over{\sqrt{(1 – x^2)}}}\\
f’(x)={dy\over{dx}} = {1\over{\sqrt{(1 – x^2)}}}
\end{matrix}\)
Properties of Arcsine
Rule name
Rule
Sine of arcsine
sin( arcsin x ) = x
Arcsine of sine
arcsin( sin x ) = x+2kπ, when k∈ℤ (k is integer)
Arcsin of negative argument
arcsin(-x) = – arcsin x
Complementary angles
arcsin x = π/2 – arccos x = 90° – arccos x
Arcsin sum
Arcsin difference
Cosine of arcsine
Tangent of arcsine
Derivative of arcsine
Indefinite integral of arcsine
Corollary
Rule name |
Rule |
Sine of arcsine |
sin( arcsin x ) = x |
Arcsine of sine |
arcsin( sin x ) = x+2kπ, when k∈ℤ (k is integer) |
Arcsin of negative argument |
arcsin(-x) = – arcsin x |
Complementary angles |
arcsin x = π/2 – arccos x = 90° – arccos x |
Arcsin sum |
|
Arcsin difference |
|
Cosine of arcsine |
|
Tangent of arcsine |
|
Derivative of arcsine |
|
Indefinite integral of arcsine |
|
Corollary |
|
Solved Examples of Derivative of Arcsin x
Example 1: What is the derivative of arcsin(x − 1)?
Solution: Derivative of inverse trigonometric functions. The general formula to differentiate the arcsin functions is
\(\begin{matrix}
\int{sin^{-1}u} = {1\over{\sqrt{(1 – u^2)}}} {du\over{dx}}\\
{d\over{dx}} sin^{-1}(x-1)={1\over{\sqrt{(1-(x-1)^2)}}}\times{d(x-1)\over{dx}}\\
{d\over{dx}} sin^{-1}(x-1)={1\over{\sqrt{(1-(x-1)^2)}}}
\end{matrix}\)
Example 2: What is the derivative of arcsin(x\a)?
Solution:
To start off, let’s set this function equal to y
\(\begin{matrix}
y=sin^{−1}({x\over{a}})\\
siny=({x\over{a}})\\
\text{ Multiply a to both sides and taking the derivative }\\
{d\over{dx}}[asiny] = {d\over{dx}}\\
{dy\over{dx}} acosy = 1\\
{dy\over{dx}} = {1\over{acosy}}\\
\text{ Divide both sides to isolate }{dy\over{dx}}\\
{dy\over{dx}} = {1\over{a}}secy\\
secy = {1\over{cosy}}\\
\text{ from the image below}\\
secy = {a\over{\sqrt{a2^−x^2}}}\\
\text{ We will now substitute this value back into the answer for our derivative: }
{dy\over{dx}} = {1\over{a}}secy\\
{dy\over{dx}} = {1\over{a}}{a\over{\sqrt{a2^−x^2}}}\\
\text{ Canceling out the a in the numerator and denominator, we are left with our final answer: }\\
{dy\over{dx}} = {1\over{\sqrt{a2^−x^2}}}
\end{matrix}\)
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FAQs For Derivative of Arcsin
What is Derivative of arcsin x?
Derivative of arcsin function is denoted by d/dx(arcsin x) and its value is 1/√1-x². It returns the angle whose sine is a given number.
What is Arcsin function?
Arcsin function is the inverse of the sine function and is a pure trigonometric function.
How do you find the derivative of a arcsin x function?
What is the derivative of arcsin of square root x?
The derivative of arcsin x is
What are the properties of arcsin x?
Sine of arcsine: sin( arcsin x ) = xArcsine of sine: arcsin( sin x ) = x+2kπ, when k∈ℤ (k is integer)Arcsin of negative argument: arcsin(-x) = - arcsin xComplementary angles: arcsin x = π/2 - arccos x = 90° - arccos xArcsin sum:
Why do we use the inverse trigonometric function arcsin(x)?
The arcsin(x) function helps us find the angle whose sine is x. It’s used in calculus, geometry, and physics where we need to reverse the sine function.
What does the graph of arcsin(x) tell us about its derivative?
The graph of arcsin(x) increases slowly as x approaches ±1, where the slope becomes steeper. This matches the derivative, which approaches infinity near x = ±1 due to the square root in the denominator.
Is the derivative of arcsin(x) continuous over its domain?
Yes, it is continuous and differentiable in the open interval (−1,1) but it becomes undefined at x = ±1.
In which real-world applications is the derivative of arcsin(x) used?
It appears in problems involving wave motion, pendulum physics, angle measurements in navigation, and inverse kinematics in robotics.
Can the Chain Rule be applied to arcsin(f(x))?
Yes. If y = arcsin(f(x)), then using the Chain Rule: dy/dx = f ′(x) / √(1 - [f(x)]²)