Derivative of arcsin(x)
du/sqrt(1-u^2)
Using the chain rule, we can find the derivative of the arcsin(x) as follows:
Let y = arcsin(x)
Then, sin(y) = x (by definition of arcsin)
Differentiating both sides with respect to x:
cos(y) * dy/dx = 1
Simplifying for dy/dx:
dy/dx = 1/cos(y)
Since we know that sin(y) = x, we can use the Pythagorean identity to find cos(y):
cos^2(y) = 1 – sin^2(y)
cos(y) = sqrt(1 – x^2)
Substituting this value into our derivative formula:
dy/dx = 1/sqrt(1 – x^2)
Therefore, the derivative of arcsin(x) is:
d/dx (arcsin(x)) = 1/sqrt(1 – x^2)
Note that this derivative is only defined for -1 < x < 1, which is the domain of arcsin(x).
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