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Derivative of Arcsine

Complete guide with formula, proof, examples, and graph.

Quick Answer

ddxarcsin⁡(x)=11−x2\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}

The derivative of \arcsin(x) is:

Proof / Derivation

Step-by-step derivation of the derivative formula.

Start with the definition of arcsine as the inverse of sine.

Lety=arcsin⁡(x),thenx=sin⁡(y)Let y = \arcsin(x), then x = \sin(y)

Differentiate both sides implicitly with respect to x.

Differentiateimplicitly:1=cos⁡(y)⋅dydxDifferentiate implicitly: 1 = \cos(y) \cdot \frac{dy}{dx}

Solve for dy/dx.

dydx=1cos⁡(y)\frac{dy}{dx} = \frac{1}{\cos(y)}

Use the Pythagorean identity to express cos(y) in terms of x.

Sincesin⁡(y)=x,cos⁡(y)=1−sin⁡2(y)=1−x2Since \sin(y) = x, \cos(y) = \sqrt{1-\sin^2(y)} = \sqrt{1-x^2}

Substitute to obtain the final derivative formula.

∴ddxarcsin⁡(x)=11−x2\therefore \frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}

Graph

Visualization of Arcsine and its derivative.

f(x) = \arcsin(x)

f(x)=arcsin⁡(x)f(x) = \arcsin(x)

f'(x) = \frac{1}{\sqrt{1-x^2}}

f′(x)=11−x2f'(x) = \frac{1}{\sqrt{1-x^2}}
Domain: (−1,1)(-1, 1)Range: [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}]

Worked Examples

Step-by-step solutions using the chain rule and other techniques.

Find: ddxarcsin⁡(2x)\frac{d}{dx}\arcsin(2x)

Solution: 21−4x2\frac{2}{\sqrt{1-4x^2}}

1.Chainrule:u=2x,u′=2Chain rule: u = 2x, u' = 2
2.=11−(2x)2⋅2=21−4x2= \frac{1}{\sqrt{1-(2x)^2}} \cdot 2 = \frac{2}{\sqrt{1-4x^2}}

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Frequently Asked Questions

What is the derivative of Arcsine?

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The derivative of arcsin⁡(x)\arcsin(x) is 11−x2\frac{1}{\sqrt{1-x^2}}. This is one of the fundamental derivatives in calculus that you should memorize.

How do you prove the derivative of Arcsine?

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The proof uses the limit definition of the derivative. See the Proof section above for the complete step-by-step derivation.

Is the derivative of Arcsine always the same?

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Yes, the derivative formula for arcsin⁡(x)\arcsin(x) is constant — it does not depend on x. However, when composed with inner functions (e.g., arcsin⁡(x)\arcsin(x) of u(x)), the chain rule applies.

Where is the derivative of Arcsine undefined?

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The derivative is undefined where the original function is not differentiable. Check the domain section for details.

Why is the derivative of Arcsine important?

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This derivative appears frequently in physics (wave motion), engineering (signal processing), economics (oscillating models), and many other fields involving periodic or growth phenomena.