Final answer:
To find the derivative of the function f(x) = (4x-4)³(6x²/4)⁴ using logarithmic differentiation, we can take the natural logarithm of both sides and apply logarithmic properties to simplify the equation. Then, we can differentiate both sides with respect to x and solve for f'(x).
Step-by-step explanation:
To find the derivative of the function f(x) = (4x-4)³(6x²/4)⁴ using logarithmic differentiation, we will use the logarithmic properties to simplify the function and then find its derivative.
Step 1: Take the natural logarithm of both sides of the equation:
ln(f(x)) = ln((4x-4)³(6x²/4)⁴)
Step 2: Apply logarithmic properties to simplify the equation:
ln(f(x)) = 3ln(4x-4) + 4ln(6x²/4)
Step 3: Differentiate both sides of the equation with respect to x:
(1/f(x))(f'(x)) = 3(1/(4x-4))(4) + 4(1/(6x²/4))(6x/2)
Step 4: Simplify the equation and solve for f'(x):
f'(x) = f(x)((3/(4x-4))(4) + (4/(6x²/4))(6x/2))