If y = x + log y, what is dy/dx?

Derivatives & Second-Order Derivatives · Question 41 of 96

Question 41

MediumOriginal question

If $y = x + \log y$ , what is $\frac{dy}{dx}$ ?

\frac{y - 1}{y}

Correct answerYour answer

\frac{y}{1 - y}

Correct answerYour answer

1 + \frac{1}{y}

Correct answerYour answer

\frac{y}{y - 1}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

Differentiate both sides: $\frac{dy}{dx} = 1 + \frac{1}{y}\frac{dy}{dx}$ .
$\frac{dy}{dx}\left(1 - \frac{1}{y}\right) = 1$ , so $\frac{dy}{dx} = \frac{y}{y - 1}$ .

Worth noting

$1 + \frac{1}{y}$ differentiates $\log y$ as $\frac{1}{y}$ and forgets the factor $\frac{dy}{dx}$ .

Ten questions from Derivatives & Second-Order Derivatives, at your own pace.

Ten questions from Derivatives & Second-Order Derivatives, at your own pace.