Let y = (x + 1)^x for x > -1. Which of the following statements are true? I. log y = xlog(x + 1). II. 1/ydy/dx = log(x + 1) + x/(x + 1). III. dy/dx = 0 at x = 0. IV. dy/dx = x(x + 1)^x - 1.

Question 95

MediumOriginal question

Let $y = (x + 1)^{x}$ for $x > -1$ . Which of the following statements are true?
I. $\log y = x\log(x + 1)$ .
II. $\frac{1}{y}\frac{dy}{dx} = \log(x + 1) + \frac{x}{x + 1}$ .
III. $\frac{dy}{dx} = 0$ at $x = 0$ .
IV. $\frac{dy}{dx} = x(x + 1)^{x - 1}$ .

All four: I, II, III and IVCorrect answerYour answerI, II and III onlyCorrect answerYour answerI and II onlyCorrect answerYour answerI, II and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

I: $\log(a^{b}) = b\log a$ . Differentiating it by the product rule gives II.
At 0: $y = 1$ and $\frac{y'}{y} = \log 1 + 0 = 0$ , so III is true. IV is the power rule, which needs a constant power: false.

Worth noting

IV treats the power $x$ as a constant and misses the $\log(x + 1)$ term.

Concept: Logarithmic differentiation
Typical timeTypical: 120 s
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The rule this testsLogarithmic differentiation

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