Find the derivative of ln(arccoshx)
use the good old chain rule:
1/arccosh(x) * 1/√(x^2-1)
ok, I got this far. Can this be simplified at all?
not that I know of.
To find the derivative of ln(arccosh(x)), we can use the chain rule. The chain rule states that if we have a composite function, f(g(x)), then its derivative is given by the product of the derivative of the outer function, f'(g(x)), and the derivative of the inner function, g'(x).
Let's break down the given function step by step:
Step 1: Identify the inner function and its derivative.
In this case, the inner function is arccosh(x). The derivative of arccosh(x) can be calculated using the inverse hyperbolic trigonometric function formula, which states that the derivative of arccosh(x) is equal to 1 divided by the square root of (x^2 - 1). Therefore, g'(x) = 1/√(x^2 - 1).
Step 2: Identify the outer function and its derivative.
The outer function is ln(x). The derivative of ln(x) is simply 1/x.
Step 3: Apply the chain rule.
Using the chain rule, we can find the derivative of ln(arccosh(x)):
d/dx [ln(arccosh(x))] = (1/arccosh(x)) * (1/√(x^2 - 1))
Therefore, the derivative of ln(arccosh(x)) is (1/arccosh(x)) * (1/√(x^2 - 1)).
Note: Remember to pay attention to the domain of the function. The function arccosh(x) is only defined for x ≥ 1, so the derivative we found here is only valid in that domain.