Andy C. answered • 07/01/18

Math/Physics Tutor

ln y = (4/5) ln x + ln cosx - ln6 - ln (x + 3)

dy/y = 4/(5x) - sinx/cosx - 1/(x+3)

Multiplies both side by the original function y:

dy = [4/(5x) - sinx/cosx - 1/(x+3)] [ x^(4/5) cos x / (6(x+3))

= 4 cosx/(30 * x^(1/5) (x+3) - x^(4/5)sinx / (6(x+3)) - x^(4/5)cosx / 6(x+3)^2

= (2/15)cosx/ (x^(1/5)(x+3) - (x^(4/5)sinx / (6(x+3)) - (x^(4/5)cosx)/(6(x+3)^2)

= (4/5)cosx(x+3) - x(sinx)(x+3) - x^(4/5)(cosx) / (6 (x^(1/5))(x+3)^2)

= (x+3) [(4/5) cosx - x sin x) ] - x^(4/5) cos x / (6 x^(1/5)(x+3)^2)

which is the same as if the quotient rule were applied