If f(x) = ((x + 1)(x + 2))/((x + 3)(x + 4)), what is f'(0)?

Question 92

MediumOriginal question

If $f(x) = \frac{(x + 1)(x + 2)}{(x + 3)(x + 4)}$ , what is $f'\left(0\right)$ ?

\frac{11}{72}

Correct answerYour answer

\frac{1}{6}

Correct answerYour answer

\frac{25}{72}

Correct answerYour answer

\frac{11}{12}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$\log y = \log(x + 1) + \log(x + 2) - \log(x + 3) - \log(x + 4)$ , so at 0, $\frac{y'}{y} = 1 + \frac{1}{2} - \frac{1}{3} - \frac{1}{4} = \frac{11}{12}$ .
$y(0) = \frac{2}{12} = \frac{1}{6}$ , so $f'(0) = \frac{1}{6} \times \frac{11}{12} = \frac{11}{72}$ .

Worth noting

$\frac{25}{72}$ adds $\frac{1}{3}$ and $\frac{1}{4}$ ; the factors below the line enter with a minus sign.

Ten questions from Chain Rule, Log Differentiation & Rate of Change, at your own pace.