Find the derivative of ln(x+(x^2-1)^(1/2)).
let y = ln [ x + (x^2 - 1)^(1/2) ]
dy/dx = 1/(x + (x^2 - 1)^(1/2) * (1 + (1/2)(x^2 - 1)^(-1/2) (2x) )
= 1/(x + √(x^2 - 10 ) * (1 + x/√(x^2 - 1) )
= 1/(x + √(x^2 - 10 ) * (√(x^2 -1) + x)/√(x^2-1)
let's multiply top and bottom by x - √(x^2 - 1) , thus rationalizing the denominator in the first part
= 1/(x + √(x^2 - 10 ) * (√(x^2 -1) + x)/√(x^2-1) * [x - √(x^2 - 1)]/[x - √(x^2 - 1)]
= (x^2 - x^2 + 1)/( (√x^2 - 1)(x^2 - x^2 + 1) )
= 1/√(x^2 - 1)
Whewww!
you can make things a little less complicated if you recognize that
ln(x+√(x^2-1)) = arccosh(x)
d/dx arccosh(x) = 1/√(x^2-1)
Reiny's excellent work shows how it's done.
To find the derivative of the function ln(x+(x^2-1)^(1/2)), we can use the chain rule. The chain rule states that if we have a composite function, like ln(f(x)), then the derivative of ln(f(x)) with respect to x is given by (1/f(x)) * f'(x).
Let's break down the steps to find the derivative:
Step 1: Identify the function inside the ln.
In this case, the function inside the ln is x + (x^2 - 1)^(1/2).
Step 2: Find the derivative of the function inside the ln.
To find the derivative of x + (x^2 - 1)^(1/2), we need to apply the chain rule.
Let's denote the function as f(x) = x + (x^2 - 1)^(1/2). Then we can rewrite the function as ln(f(x)).
Now, let's find the derivative of f(x):
The derivative of x with respect to x is 1, and the derivative of (x^2 - 1)^(1/2) with respect to x can be found using the chain rule again.
Let's denote g(x) = x^2 - 1.
Then g'(x) = 2x.
Now, we have (x^2 - 1)^(1/2), let's use the chain rule:
d/dx[(x^2 - 1)^(1/2)] = (1/2)(x^2 - 1)^(-1/2) * d/dx[x^2 - 1]
= (1/2)(x^2 - 1)^(-1/2) * 2x
= x / (x^2 - 1)^(1/2)
So, f'(x) = 1 + x / (x^2 - 1)^(1/2)
Step 3: Apply the chain rule to find the derivative of ln(f(x)).
Using the chain rule, the derivative of ln(f(x)) with respect to x is (1/f(x)) * f'(x):
d/dx[ln(x + (x^2 - 1)^(1/2))] = (1 / (x + (x^2 - 1)^(1/2))) * (1 + x / (x^2 - 1)^(1/2))
Therefore, the derivative of ln(x + (x^2 - 1)^(1/2)) is:
(1 / (x + (x^2 - 1)^(1/2))) * (1 + x / (x^2 - 1)^(1/2))