# Let f(x)= ((x^9)*((x-2)^7))/(((x^2)+2)^8) Use logarithmic differentiation to determine the derivative. f'(x)= ?

## Expert Answers You should use the following rules to differentiate the function, such that: product rule, quotient rule, power rule and chain rule.

`f'(x) = (((x^9)*((x-2)^7))'*((x^2+2)^8) - ((x^9)*((x-2)^7))*(((x^2)+2)^8) ')/((x^2+2)^16)`

`((x^9)*(x-2)^7))' = 9x^8*(x-2)^7 + 7x^9*(x-2)^6`

`((x^9)*(x-2)^7))' = x^8*(x-2)^6*(9x - 18 + 7x)`

`((x^9)*(x-2)^7))' = x^8*(x-2)^6*(16x - 18)`

`f'(x) = ((x^8*(x-2)^6*(16x - 18))*((x^2+2)^8) - 14x^10*(x-2)^7*(x^2 + 2)^7)/((x^2+2)^16)`

`f'(x) = (x^8*(x-2)^6*(x^2 + 2)^7*((16x - 18)*(x^2+2) - 14x^2*(x-2)))/((x^2+2)^16)`

`f'(x) = (x^8*(x-2)^6*((16x - 18)*(x^2+2) - 14x^2*(x-2)))/((x^2+2)^9)`

Hence, evaluating the derivative of function yields `f'(x) = (x^8*(x-2)^6*((16x - 18)*(x^2+2) - 14x^2*(x-2)))/((x^2+2)^9).`

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