Step-by-step explanation:
W(x) = (10x⁴ − 8) (30x + 25)^0.5
A) Take log of both sides.
ln(W) = ln[(10x⁴ − 8) (30x + 25)^0.5]
ln(W) = ln(10x⁴ − 8) + ln[(30x + 25)^0.5]
ln(W) = ln(10x⁴ − 8) + 0.5 ln(30x + 25)
Take derivative.
W' / W = 40x³ / (10x⁴ − 8) + 0.5 (30) / (30x + 25)
W' / W = 20x³ / (5x⁴ − 4) + 3 / (6x + 5)
W' = W [20x³ / (5x⁴ − 4) + 3 / (6x + 5)]
W'(x) = (10x⁴ − 8) (30x + 25)^0.5 [20x³ / (5x⁴ − 4) + 3 / (6x + 5)]
B) Evaluate at x = 0.
W'(0) = (0 − 8) (0 + 25)^0.5 [0 / (0 − 4) + 3 / (0 + 5)]
W'(0) = (-8) (5) (0 + 3/5)
W'(0) = -24
