Discover How To Find The Derivative Of Arcsin X Using Implicit Differentiation

derivative of sin^-1

1/√(1-x^2)

The derivative of sin^-1(x) (also expressed as arcsin x or inverse sine of x) can be found using implicit differentiation. Let y = sin^-1(x). Then, taking the sine of both sides of the equation y = sin^-1(x), we get:

sin(y) = x

Now, we can differentiate both sides of this equation with respect to x using the chain rule:

cos(y) * dy/dx = 1

Solving for dy/dx, we have:

dy/dx = 1/cos(y)

Using the identity sin^2(y) + cos^2(y) = 1, we can solve for cos(y):

cos(y) = √(1 – sin^2(y))

Substituting this expression into our equation for dy/dx, we get:

dy/dx = 1/√(1 – sin^2(y))

Finally, substituting our original expression y = sin^-1(x), we have:

dy/dx = 1/√(1 – x^2)

Therefore, the derivative of sin^-1(x) is:

d/dx(sin^-1(x)) = 1/√(1 – x^2)

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