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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