f=(x)cosh^(-1)((x)/(2))-√((x)^(2)-4)
f'=
To find the derivative of the function f(x), we need to take the derivative of each term separately. Let's break down the given function into its individual components:
f(x) = (x)cosh^(-1)((x)/(2)) - √((x)^(2) - 4)
1. Derivative of (x)cosh^(-1)((x)/(2)):
To find the derivative of this term, we can apply the chain rule. The chain rule states that if we have a composition of functions, the derivative will be the derivative of the outer function multiplied by the derivative of the inner function. In this case, the outer function is cosh^(-1)((x)/(2)) and the inner function is (x). Let's find the derivatives of these functions individually.
Derivative of cosh^(-1)((x)/(2)):
To find the derivative of cosh^(-1)((x)/(2)), we can use the inverse hyperbolic cosine derivative formula, which states that d/dx(cosh^(-1)(u)) = 1 / sqrt(u^2 - 1), where u = (x)/(2). Let's substitute u with (x)/(2) in the formula:
d/dx(cosh^(-1)((x)/(2))) = 1 / sqrt(((x)/(2))^2 - 1)
= 1 / sqrt((x^2)/(4) - 1)
= 1 / sqrt((x^2 - 4) / 4) [common denominator]
= 1 / (sqrt(x^2 - 4) / 2)
= 2 / sqrt(x^2 - 4)
Derivative of (x):
The derivative of (x) with respect to x is simply 1.
Multiplying these two derivatives together, we get:
(x)cosh^(-1)((x)/(2)) = x * (2 / sqrt(x^2 - 4))
= (2x) / sqrt(x^2 - 4)
2. Derivative of √((x)^(2) - 4):
To find the derivative of this term, we can apply the power rule. The power rule states that if we have a term of the form sqrt(u), its derivative would be (1/2) * u^(-1/2) * u', where u = (x)^(2) - 4. Let's find the derivative:
Derivative of √((x)^(2) - 4):
Using the power rule, we have:
d/dx(√((x)^(2) - 4)) = (1/2) * ((x)^(2) - 4)^(-1/2) * (2x)
= x / sqrt((x)^(2) - 4)
Now that we have the derivatives of each term:
(2x) / sqrt(x^2 - 4) - x / sqrt(x^2 - 4)
We can combine these derivatives into one expression:
f'(x) = (2x - x) / sqrt(x^2 - 4)
= x / sqrt(x^2 - 4)
Therefore, the derivative of f(x) is f'(x) = x / sqrt(x^2 - 4).