If
f(x) = x⁶ (x - 8)⁷ / (x² + 2)²
then taking the logarithm of both sides lets us expand the right side as
ln(f(x)) = ln(x⁶ (x - 8)⁷ / (x² + 2)²)
ln(f(x)) = ln(x⁶) + ln((x - 8)⁷) - ln((x² + 2)²)
using the property ln(ab) = ln(a) + ln(b).
We can also use the property ln(aⁿ) = n ln(a), so that
ln(f(x)) = 6 ln(x) + 7 ln(x - 8) - 2 ln(x² + 2)
Then differentiating both sides using the chain rule gives
f'(x)/f(x) = 6 x'/x + 7 (x - 8)'/(x - 8) - 2 (x² + 2)'/(x² + 2)
f'(x)/f(x) = 6 (1)/x + 7 (1)/(x - 8) - 2 (2x)/(x² + 2)
f'(x)/f(x) = 6/x + 7/(x - 8) - 4x/(x² + 2)
Solve for f'(x) :
f'(x) = (6/x + 7/(x - 8) - 4x/(x² + 2)) f(x)
and replace f(x) with the given function :
f'(x) = (6/x + 7/(x - 8) - 4x/(x² + 2)) • x⁶ (x - 8)⁷ / (x² + 2)²
We can expand the product :
f'(x) = 6 x⁵ (x - 8)⁷ / (x² + 2)²+ 7x⁶ (x - 8)⁶ / (x² + 2)² - 4x⁷ (x - 8)⁷ / (x² + 2)³
then simplify by factoring :
f'(x) = x⁵ (x - 8)⁶/(x² + 2)³ • (6 (x - 8) (x² + 2) + 7x (x² + 2) - 4x² (x - 8))
Simplify the rest :
f'(x) = x⁵ (x - 8)⁶ (9x³ - 16x² + 26x - 96)/(x² + 2)³
