Final answer:
To find f'(x) using logarithmic differentiation, take the natural logarithm of both sides of the equation and simplify using logarithmic properties. Then, differentiate both sides with respect to x.
Step-by-step explanation:
To find f'(x) using logarithmic differentiation, we can take the natural logarithm of both sides of the equation y = (2x - 5)³ (9 - x)⁵ / (4x³ - 6)⁹. This allows us to simplify the equation and differentiate it using logarithmic properties.
First, we take the natural logarithm of both sides: ln(y) = ln((2x - 5)³ (9 - x)⁵ / (4x³ - 6)⁹).
Next, we use logarithmic properties to simplify the equation and differentiate: ln(y) = 3ln(2x - 5) + 5 ln(9 - x) - 9ln(4x³ - 6). Then, differentiate both sides of the equation with respect to x. The derivatives of ln(y) and ln(x) can be found using the chain rule, and we solve for f'(x).