Use logarithmic differentiation to determine the derivative.

Question

Let f(x)=(x^2(x-8)^6)/((x^2+6)^7)

Use logarithmic differentiation to determine the derivative.

Answer 1

y = (x^2)(x-8)^6 / (x^2+6)^7

log y = 2logx + 6 log (x-8) - 7 log (x^2+6)

y'/y = 2/x + 6/(x-8) - 7(2x)/(2x+6)

= 2/x + 6/(x-8) - 7/(x+3)

y' = y [ 2/x + 6/(x-8) - 7/(x+3)]

Distributing the original function:

first term: x cancels but 2 survives

2nd term: (x-8) cancels but 6 survives

3rd term: nothing happens; 7/(x+3) survives

the common denominator is (x+3)(x^2+6)^7

the derivative is:

{(x+3)[2x(x-8)^6 + 6x^2(x-8)^5] - 7x^2(x-8)^6} / (x^2+6)^7(x+3)