Derivative of arcsin(x)
du/sqrt(1-u^2)
To find the derivative of arcsin(x), we can use the formula for the derivative of an inverse function, which says:
(d/dx) [f^-1(x)] = 1 / [f'(f^-1(x))]
Let’s use this formula with f(x) = sin(x) and f^-1(x) = arcsin(x):
(d/dx) [arcsin(x)] = 1 / [sin'(arcsin(x))]
Now, we need to find the derivative of sin(x) and substitute the inverse function arcsin(x). The derivative of sin(x) is cos(x), so:
(d/dx) [arcsin(x)] = 1 / [cos(arcsin(x))]
To simplify further, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to find cos(x) in terms of sin(x):
cos(x) = sqrt(1 – sin^2(x))
Substituting sin(x) with x and arcsin(x) with θ, we get:
cos(θ) = sqrt(1 – x^2)
Therefore, the derivative of arcsin(x) is:
(d/dx) [arcsin(x)] = 1 / [sqrt(1 – x^2)]
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