Learn How To Find The Derivative Of The Inverse Hyperbolic Cosecant Function

d/dx (csch^-1 x)

– 1 / | x | √(x^2 + 1)

We can start by using the definition of the hyperbolic cosecant inverse function:

csch^-1 x = ln (x + sqrt(x^2 + 1))

Then, we can use the chain rule to find the derivative with respect to x:

d/dx (csch^-1 x) = d/dx [ln (x + sqrt(x^2 + 1))]

Using the chain rule, we can write this as:

d/dx [ln u] = 1/u * du/dx

where u = x + sqrt(x^2 + 1).

Taking the derivative of u with respect to x, we have

du/dx = 1 + (x / sqrt(x^2 + 1))

Therefore,

d/dx (csch^-1 x) = 1 / (x + sqrt(x^2 + 1)) * (1 + x / sqrt(x^2 + 1))

Simplifying this expression gives:

d/dx (csch^-1 x) = 1 / (x^2 + sqrt(x^2 + 1) * x)

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