Finding the Derivative of the Inverse Cosecant Function | Calculus Tutorial

csc^-1(x) derivative

The derivative of the inverse cosecant function, denoted as csc^(-1)(x) or arcsin(1/x), can be found using calculus

The derivative of the inverse cosecant function, denoted as csc^(-1)(x) or arcsin(1/x), can be found using calculus.

Let’s start by using the definition of the inverse function:

y = csc^(-1)(x) or arcsin(1/x)

To find the derivative of y with respect to x, we can take the derivative of both sides of the equation. However, since csc^(-1)(x) is not an elementary function, we need to use some trigonometric identities to simplify the expression.

Recall that the cosecant function (csc(x)) is equal to 1/sin(x). Using this relationship, we can rewrite the equation as:

x = 1/sin(y)

Now, we can take the derivative of both sides with respect to x using the chain rule:

1 = (d/dx) [1/sin(y)]

To simplify further, let’s use the quotient rule:

1 = (sin(y)(d/dx)(1) – 1(d/dx)(sin(y)))/(sin^2(y))

Since (d/dx)(1) is 0 (since 1 is a constant), we can simplify the expression to:

1 = – (d/dx)(sin(y))/(sin^2(y))

Now, we just need to find the derivative of sin(y) with respect to x. Applying the chain rule once again gives:

1 = – (d/dx)(sin(y))/(sin^2(y)) = -(cos(y))/(sin^2(y)) * (dy/dx)

To isolate dy/dx, we can rearrange the equation:

dy/dx = -sin^2(y)/cos(y)

However, y is equal to csc^(-1)(x) or arcsin(1/x). Substituting y back into the equation gives the final answer:

dy/dx = -sin^2(csc^(-1)(x))/cos(csc^(-1)(x))

Note that this expression can also be written as:

dy/dx = -1/(|x| * sqrt(x^2 – 1))

So, the derivative of csc^(-1)(x) is -1/(|x| * sqrt(x^2 – 1)).

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