calculate the derivative of g(x) using logarithmic differentiation.
We start by taking the natural logarithm of both sides of the equation:
ln(g(x)) = ln(x²(x - 1)³ / (x + 2)³(x² + 1)³)
Next, we can apply the logarithmic differentiation rule, which states that the derivative of ln(f(x)) is f'(x) / f(x):
ln(g(x)) = ln(x²) + ln((x - 1)³) - ln((x + 2)³) - ln((x² + 1)³)
Now, let's differentiate both sides with respect to x:
(g'(x) / g(x)) = (2x / x²) + (3(x - 1) / (x - 1)) - (3(x + 2) / (x + 2)) - (3(x² + 1) / (x² + 1))
Simplifying the expression:
g'(x) / g(x) = (2 / x) + 3 - 3(x + 2) / (x + 2) - 3(x² + 1) / (x² + 1)
Finally, we can multiply both sides by g(x) to isolate g'(x):
g'(x) = g(x) * [(2 / x) + 3 - 3(x + 2) / (x + 2) - 3(x² + 1) / (x² + 1)]
That's the derivative of g(x) using logarithmic differentiation!