Derivative of Arcsin x – Formula & Proof using Quotient Rule, Chain Rule, and First Principle

Last Updated on Jun 12, 2025
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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

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:

  1. Proof of Derivative of Arcsin by Quotient Rule
  2. Proof of Derivative of Arcsin by first principle of derivative
  3. 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 and the hypotenuse is 1. Thus, which means

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

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
\(\begin{matrix}
y=sin^{−1}({x\over{a}})\\
siny=({x\over{a}})\\
\text{ Multiply a to both sides and taking the derivative }\\
{d\over{dx}}[asiny] = {d\over{dx}}\\
{dy\over{dx}} acosy = 1\\
{dy\over{dx}} = {1\over{acosy}}\\
\text{ Divide both sides to isolate }{dy\over{dx}}\\
{dy\over{dx}} = {1\over{a}}secy\\
secy = {1\over{cosy}}\\
\text{ from the image below}\\
secy = {a\over{\sqrt{a2^−x^2}}}\\
\text{ We will now substitute this value back into the answer for our derivative: }
{dy\over{dx}} = {1\over{a}}secy\\
{dy\over{dx}} = {1\over{a}}{a\over{\sqrt{a2^−x^2}}}\\
\text{ Canceling out the a in the numerator and denominator, we are left with our final answer: }\\
{dy\over{dx}} = {1\over{\sqrt{a2^−x^2}}}
\end{matrix}\)

If you are checking Derivative of Arcsin x article, also check the related maths articles:

Applications of Derivatives

Partial Derivative

Basics of Derivatives

Derivative Rules

Derivative of x

Derivative of cot x

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FAQs For Derivative of Arcsin

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.

Arcsin function is the inverse of the sine function and is a pure trigonometric function.

The derivative of arcsin x is . By using this formula and chain rule, we can find the derivative of .

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: Arcsin difference: Cosine of arcsine: Tangent of arcsine: Derivative of arcsine: Indefinite integral of arcsine: Corollary:

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.

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.

Yes, it is continuous and differentiable in the open interval (−1,1) but it becomes undefined at x = ±1.

It appears in problems involving wave motion, pendulum physics, angle measurements in navigation, and inverse kinematics in robotics.

Yes. If y = arcsin(f(x)), then using the Chain Rule: dy/dx = f ′(x) / √(1 - [f(x)]²)

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