Derivative of Arcsin X
If there is an arcsin function X, then the derivative of arcsin is given by dividing one by the square root of the value obtained from subtracting the square of x from one. This is shown as a formula below:
Derivative of ArcSin = 1/square root of (1-square of X)
Here X belongs to real number in such a way that its value lies between -1 and +1.
Let us prove the formula for derivative of arc-sin X. If Y = arcsin X where X is a real number lying between -1 and +1, then we can say that X = sin Y. By using the derivative of sine function, we can rewrite in the derivative form as dX/dY = cos Y. By applying the derivative of inverse function, it becomes dY/dX = 1/cos Y. We know that cos square Y + sin square Y is equal to 1. We can rewrite it as cos Y equal to plus or minus of the square root of one minus sin square Y. As cos Y is greater than or equal to Zero on the arcsin X range that ranges from –pi/2 to +pi/2, we can say that cos Y always takes a positive value. So we get the value of dy/dx as one divided by the square root of the value obtained from subtracting sin square y from one. In other words, the differentiation or the derivative of arc-sin X is equal to one divided by the square root of the value obtained from subtracting the square of X from one, as we know X = sin Y. Thus, Derivative of Arc-Sin = 1/square root of (1-square of X)
Derivative of Arcsin X 2
As per the chain rule, the derivative of a square of any function is given by twice the function itself. By applying this rule, we can find out the derivative of arc-sin X2 from the formula of arcsin X.
We have seen that g(X) = arcsin X. We know the differentiation of arcsin X i.e. d/dx arcsin(X) is obtained from the formula 1/square root of (1-square of X). By applying the chain rule, we get 2arcsin(X) multiplied with (1/square root of (1-square of X)). Thus, Derivative of arc-sin X2 = 2arcsin(X)/square root (1-square of X).
Derivative of arcsin(X)+arcos(X)
The differentiation of the summation of arcsin(y) and the arcos(y) is given by pi divided by 2, i.e. arcsin(X) +arccos(X) = pi/2.
Applications of Arcsin
An important application of the arcsin transformation can be seen in the analysis of binary data.
