William W. answered • 07/28/21
Tutor
4.9
(1,021)
Experienced Tutor and Retired Engineer
y = (x4 + 2)2(x5 + 4)4
ln(y) = ln[(x4 + 2)2(x5 + 4)4]
ln(y) = ln[(x4 + 2)2] + ln[(x5 + 4)4]
ln(y) = 2ln(x4 + 2) + 4ln(x5 + 4)
d(ln(y)) = d{2ln(x4 + 2) + 4ln(x5 + 4)}
dx dx
(1/y)•(dy/dx) = [2/(x4 + 2)]•(4x3) + [4/(x5 + 4)]•(5x4)
(1/y)•(dy/dx) = 8x3/(x4 + 2) + 20x4/(x5 + 4)
dy/dx = y[8x3/(x4 + 2) + 20x4/(x5 + 4)]
dy/dx = (x4 + 2)2(x5 + 4)4[8x3/(x4 + 2) + 20x4/(x5 + 4)]
dy/dx = 8x3(x4 + 2)(x5 + 4)4 + 20x4(x4 + 2)2(x5 + 4)3
dy/dx = 4x3(x4 + 2)(x5 + 4)3[2(x5 + 4) + 5x(x4 + 2)]
dy/dx = 4x3(x4 + 2)(x5 + 4)3[2x5 + 8 + 5x5 + 10x]
dy/dx = 4x3(x4 + 2)(x5 + 4)3(7x5 + 10x + 8)
