Differentiate the function.
h(x)= ln(x+ sqrt(x^2 -4))
1/sqrt(x^2 - 4)
h'(x) = (1/(x + (x^2 - 4)^(1/2)) ( 1 + 2x(x^2 - 4)^(-1/2))
= (1 + 2x/√(x^2-4)) )/(1 + √(x^2-4) )
To differentiate the function h(x) = ln(x + √(x^2 - 4)), we can use the chain rule of differentiation. The chain rule states that when we have a composition of functions, the derivative is given by the derivative of the outer function multiplied by the derivative of the inner function.
Let's break down the function h(x) into its two components:
Outer function: ln(u)
Inner function: u = x + √(x^2 - 4)
Now, we need to find the derivatives of both the outer and inner functions:
Derivative of the outer function ln(u):
The derivative of ln(u) with respect to u is 1/u.
Derivative of the inner function u = x + √(x^2 - 4):
To differentiate this function, we need to apply the chain rule.
Let's calculate the derivative step by step:
Derivative of x with respect to x is 1.
Derivative of √(x^2 - 4) with respect to x:
To find the derivative of √(x^2 - 4), we can rewrite it as (x^2 - 4)^0.5 and apply the power rule of differentiation.
The power rule states that if we have a function of the form f(x) = x^n, the derivative is given by f'(x) = n * x^(n-1).
So, the derivative of (x^2 - 4)^0.5 is 0.5 * (x^2 - 4)^(-0.5).
Now that we have the derivative of the inner function, we can find the derivative of the outer function:
Derivative of ln(u) = 1/u multiplied by the derivative of the inner function d(u)/dx.
d(u)/dx = d/dx (x + √(x^2 - 4))
= 1 + 0.5 * (x^2 - 4)^(-0.5) * (2x) (using the chain rule)
Simplifying the expression, we have:
d(u)/dx = 1 + x / √(x^2 - 4)
Finally, multiplying the derivative of the outer function (1/u) by the derivative of the inner function (1 + x / √(x^2 - 4)), we get:
h'(x) = (1 + x / √(x^2 - 4)) / (x + √(x^2 - 4))
Therefore, the derivative of the function h(x) = ln(x + √(x^2 - 4)) is h'(x) = (1 + x / √(x^2 - 4)) / (x + √(x^2 - 4)).