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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