find f' of f=(x)cosh^(-1)((x)/(2))-√((x)^(2)-4)
To find the derivative of the given function f(x) = (x)cosh^(-1)((x)/(2)) - √((x)^(2) - 4), we can differentiate each term separately using the chain rule and the power rule.
Let's break down the function into its two terms:
f(x) = (x)cosh^(-1)((x)/(2)) - √((x)^(2) - 4)
Term 1: (x)cosh^(-1)((x)/(2))
Term 2: -√((x)^(2) - 4)
Now, let's find the derivative of each term.
For Term 1: (x)cosh^(-1)((x)/(2))
We will use the chain rule. Let u = (x)/(2).
So, f_1(x) = u*cosh^(-1)(u)
Using the chain rule, the derivative of f_1 can be found as follows:
f_1'(x) = (d/dx)[u*cosh^(-1)(u)]
= (d/dx)[u] * cosh^(-1)(u) + u * (d/dx)[cosh^(-1)(u)]
= (d/dx)[(x)/(2)] * cosh^(-1)((x)/(2)) + (x)/(2) * (d/dx)[cosh^(-1)((x)/(2))]
To find (d/dx)[(x)/(2)], we use the power rule:
(d/dx)[(x)/(2)] = (1/2) * (d/dx)[x]
= (1/2) * 1
= 1/2
To find (d/dx)[cosh^(-1)((x)/(2))], we use the chain rule again:
(d/dx)[cosh^(-1)((x)/(2))] = (d/du)[cosh^(-1)(u)] * (d/dx)[((x)/(2))]
= (d/du)[cosh^(-1)(u)] * (1/2)
Now, for Term 2: -√((x)^(2) - 4)
We will use the power rule. Let v = (x)^(2) - 4.
So, f_2(x) = -√(v)
Using the power rule, the derivative of f_2 can be found as follows:
f_2'(x) = (d/dx)[-√(v)]
= -1/2 * (d/dx)[v^(-1/2)]
To find (d/dx)[v^(-1/2)], we use the chain rule:
(d/dx)[v^(-1/2)] = (d/du)[u^(-1/2)] * (d/dx)[(x)^(2) - 4]
= (-1/2) * (d/dx)[(x)^(2) - 4]
Now we can put it all together:
f'(x) = f_1'(x) + f_2'(x)
= [(1/2) * cosh^(-1)((x)/(2))] + [-1/2 * ((x)^(2) - 4)]
Therefore, the derivative of the function f(x) = (x)cosh^(-1)((x)/(2)) - √((x)^(2) - 4) is:
f'(x) = (1/2) * cosh^(-1)((x)/(2)) - 1/2 * ((x)^(2) - 4)