Evaluating the derivative of arccsc(x) using the chain rule and inverse trigonometric function derivatives

Derivative of arccsc(x)

To find the derivative of the function arccsc(x), we can use the chain rule and the derivative of the inverse trigonometric function

To find the derivative of the function arccsc(x), we can use the chain rule and the derivative of the inverse trigonometric function.

First, let’s rewrite arccsc(x) in terms of the arcsine function:

arccsc(x) = arcsin(1/x)

Now, applying the chain rule, we differentiate the arcsin(1/x) function with respect to x:

d/dx [arcsin(1/x)] = d/dx [arcsin(u)]

where u = 1/x.

To find this derivative, we use the formula for the derivative of the arcsine function:

d/dx [arcsin(u)] = 1 / sqrt(1 – u^2) * du/dx

where du/dx is the derivative of u with respect to x.

Substituting back u = 1/x, we have:

d/dx [arcsin(1/x)] = 1 / sqrt(1 – (1/x)^2) * d/dx [1/x]

Simplifying further, we get:

d/dx [arcsin(1/x)] = 1 / sqrt(1 – 1/x^2) * (-1/x^2)

Now, combining terms, we have:

d/dx [arcsin(1/x)] = -1 / (x^2 * sqrt(1 – 1/x^2))

Therefore, the derivative of arccsc(x) with respect to x is:

d/dx [arccsc(x)] = -1 / (x^2 * sqrt(1 – 1/x^2))

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