Does the derivative of f(x)=log_7((x-2)^2sqrt(x+6)) equal (5x+22)/2ln(7)(x-2)(x+6)

Answer 1

Answer 2

f(x) = log₇((x-2)²√(x+6)) = (1/ln(7))ln((x-2)²√(x+6))

Ignore 1/ln(7) for now

Taking the derivative, using the chain rule (ln(u))' = u'/u

(1/((x-2)²√(x+6)) ((2x-4)√(x+6) + (x-2)²/2√(x+6))

= (1/((x-2)²√(x+6))[(2x-4)(x+6) + (x-2)²)/2√(x+6))

= (4x²+16x-48+x²-4x+4)/(2(x-2)²(x+6))

= (5x²+12x-44)/(2(x-2)²(x+6)) = ((5x+22)(x-2))/(2(x-2)²(x+6))

= (5x+22)/(2(x-2)(x+6))

f'(x) = (5x+22)/(2ln(7)(x-2)(x+6))

So the answer is yes