How to Find the Derivative of Arccsc(x) using Chain Rule and Trigonometric Identities

Derivative of arccsc(x)

-du/abs(u)sqrt(u^2-1)

Let y = arccsc(x), then csc(y) = x and y lies in the interval [0, π].

Now, we can use the chain rule to differentiate both sides with respect to x as follows:

d/dx [csc(y)] = d/dx [x]

Applying the chain rule, we get:

-csc(y)cot(y)dy/dx = 1

dy/dx = -cot(y)/csc(y)

To find the value of cot(y) and csc(y), we can use the identity:

csc^2(y) – 1 = cot^2(y)

Since csc(y) = x and y lies in the interval [0, π], we have:

sin(y) = 1/x and cos(y) = sqrt(1 – sin^2(y)) = sqrt(1 – 1/x^2)

Therefore, cot(y) = cos(y)/sin(y) = sqrt(1 – 1/x^2)/1/x = x/sqrt(x^2 – 1)

And csc(y) = x

Substituting these values into the expression above, we get:

dy/dx = -sqrt(x^2 – 1)/x^2

Therefore, the derivative of arccsc(x) is -sqrt(x^2 – 1)/x^2.

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