Let f(x) = ln(((2x–5) / (7x+4))^(1/2)). f'(x) = ...

I keep on getting the wrong answer.

To find the derivative of the given function f(x), you can use the chain rule and the power rule for differentiation.

Step 1: Rewrite the function using exponentiation rules:
f(x) = ln(((2x–5) / (7x+4))^(1/2)) = ln(((2x–5) / (7x+4))^0.5)

Step 2: Apply the chain rule. For a composite function u(v(x)), the derivative can be expressed as d(u(v))/dx = du/dv * dv/dx. In this case, let's define u = ln(v) and v = ((2x–5) / (7x+4))^0.5.

First, find the derivative of u(v):
du/dv = 1/v

Then, find the derivative of v(x):
Let's differentiate v by rewriting ((2x–5) / (7x+4))^0.5 as a power function:
v = (2x–5) / (7x+4))^0.5 = ((2x–5) / (7x+4))^0.5

To differentiate v, apply the power rule:
dv/dx = 0.5 * ((2x–5) / (7x+4))^(-0.5) * d((2x–5) / (7x+4))/dx

Step 3: Differentiate d((2x–5) / (7x+4))/dx using the quotient rule:
Let's define u = (2x–5) and v = (7x+4). Applying the quotient rule, we have:
d(u/v)/dx = (v * du/dx - u * dv/dx) / v^2

Substituting u = (2x–5) and v = (7x+4):
d((2x–5) / (7x+4))/dx = ((7x+4) * (2) - (2x–5) * (7)) / (7x+4)^2

Step 4: Substitute the derivative of u/v and dv/dx into the expression dv/dx obtained in Step 2:
dv/dx = 0.5 * ((2x–5) / (7x+4))^(-0.5) * ((7x+4) * 2 - (2x–5) * 7) / (7x+4)^2

Step 5: Multiply du/dv and dv/dx obtained in Steps 2 and 4, respectively:
f'(x) = du/dv * dv/dx
f'(x) = (1/v) * (0.5 * ((2x–5) / (7x+4))^(-0.5) * ((7x+4) * 2 - (2x–5) * 7) / (7x+4)^2)

Simplifying the expression further should give you the correct derivative of f(x). Make sure to double-check your calculations to avoid any arithmetic errors.