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) $2\ln(x)$

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) $2x^x(\ln(x)+1)$

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