Test your understanding of the derivative of \( x^x \), logarithmic differentiation, and related logarithm properties.

Try to answer the questions before checking the answers at the bottom of the page.

Question 1

What is:

\( \frac{d}{dx}​(x^x)\)

A) $x^{x-1}$
B) $x^x\ln(x)$
C) $x^x(\ln(x)+1)$
D) $x^{x+1}$

Question 2

Which logarithm property is used to simplify $\ln(x^x)$?

A) $\ln(a+b)=\ln(a)+\ln(b)$
B) $\ln(a^b)=b\ln(a)$
C) $\ln(ab)=\ln(a)\ln(b)$
D) $\ln(a-b)=\ln(a)-\ln(b)$

Question 3

What is:

\( ln(1) \)

A) 0
B) 1
C) $e$
D) Undefined

Question 4

Find:

\( \frac{d}{dx}​(x^x)|_{x=1} \)​

A) 0
B) 1
C) 2
D) $e$

Question 5

Find:

\( \frac{d}{dx}​(x^x)|_{x=e} \)​

A) $e^e$
B) $2e^e$
C) $e$
D) $2e$

Question 6

Find:

\( \frac{d}{dx}​(2 x^x) \)

A) $2x^x$
B) $x^x(\ln(x)+1)$
C) $2x^x(\ln(x)+1)$
D) $2\ln(x)$

Question 7

Which expression is equivalent to:
\(x^x\)

A) \(e^{x+\ln(x)}\)

B) \(e^{x\ln(x)}\)

C) \(e^{\ln(x)}\)

D) \(xe^x\)

Question 8

What is:

\(\frac{d}{dx}\bigl(x\ln(x)\bigr)\)

A) \(\ln(x)\)

B) \(1\)

C) \(\ln(x)+1\)

D) \(x\ln(x)+1\)

Question 9

Using logarithmic differentiation, if:

\(y=x^x\)

what do we get after differentiating:

\(\ln(y)=x\ln(x)\)

A) \(y’=\ln(x)+1\)

B) \(\frac{y’}{y}=\ln(x)+1\)

C) \(\frac{y}{y’}=\ln(x)+1\)

D) \(y’=x\ln(x)\)

Question 10

At which positive value of (x) is:

\(\frac{d}{dx}\left(x^x\right)=0\)

A) \(x=1\)

B) \(x=e\)

C) \(x=\frac{1}{e}\)

D) \(x=0\)

Answers

1. C) $x^x(\ln(x)+1)$
2. B) $\ln(a^b)=b\ln(a)$
3. A) 0
4. B) 1
5. B) $2e^e$
6. C) $2x^x(\ln(x)+1)$
7. B) \(e^{x\ln(x)}\)
8. C) \(\ln(x)+1\)
9. B) \(\frac{y’}{y}=\ln(x)+1\)
10. C) \(x=\frac{1}{e}\)

Practice

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