Differentiate f. f(x)= (x)/1-ln(x-6)?

Question

f(x)= (x)/1-ln(x-6)

f(x)=

Find the domain of f

Answer 1

f(x) = x/[1 - log(x - 6)]

Apply 'derivative with respect to x' each side.

f'(x) = (d/dx){x/[1 - log(x - 6)]}

The Quotient Rule: (U/V)' = [VU' - V'U]/V²

Here u = x ⇒ u' = 1.

v = [1 - log(x - 6)] ⇒ v' = 0 - [1/(x - 6)](1-0) ⇒ v' = -1/(x - 6)

f'(x) = {[1 - log(x - 6)](1) - [-1/(x - 6)]x} / [1 - log(x - 6)]²

f'(x) = [1 - log(x - 6) + x/(x - 6)]/ [1 - log(x - 6)]²

Rewrite the expression with common denominator.

f'(x) = [x - 6 - (x - 6)log(x - 6) + x] / (x - 6)[1 - log(x - 6)]²

f'(x) = [2x - 6 - (x - 6)log(x - 6)]/(x - 6)[1 - log(x - 6)]²

Recall:  Essentially the domain of some f(x) refers to the x values it may exist in. For instance, the domain of sine is the set of all real numbers, R, whereas the domain of the square root (x) consists only of numbers >= 0

The domain of 'f' to be defined for all positive real numbers.