Derivative of arcsin(x)
The derivative of the arcsin(x) function can be found using the chain rule
The derivative of the arcsin(x) function can be found using the chain rule. To begin, let’s assume y = arcsin(x). Our goal is to find dy/dx.
The inverse of the arcsin function is the sine function, so we can rewrite our equation as x = sin(y).
Now, we take the derivative of both sides with respect to x:
d/dx(x) = d/dx(sin(y))
Since x is a variable with respect to x, the derivative of x with respect to x is simply 1. The derivative of sin(y) with respect to x can be found using the chain rule, which states that if u = f(g(x)), then du/dx = f'(g(x)) * g'(x).
In our case, f(u) = sin(u) and g(x) = y. Applying the chain rule, we have:
1 = cos(y) * dy/dx
Rearranging the equation, we get:
dy/dx = 1/cos(y)
Since x = sin(y) and y = arcsin(x), we can substitute y = arcsin(x) into the equation:
dy/dx = 1/cos(arcsin(x))
To simplify further, we recall that cos(arcsin(x)) can be found using the Pythagorean identity:
cos(arcsin(x)) = √(1 – sin^2(arcsin(x)))
Since sin(arcsin(x)) simplifies to just x, we have:
cos(arcsin(x)) = √(1 – x^2)
Now, substituting back into the equation:
dy/dx = 1/√(1 – x^2)
Therefore, the derivative of arcsin(x) with respect to x is 1/√(1 – x^2).
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