In this page we practice finding the derivative of \( x^x \) using logarithmic differentiation and the derivative formula:
\( \frac{d}{dx}(x^x) = x^x (ln(x)+1) \)
Example 1
Find:
\( \frac{d}{dx}(x^x) \)
Use the formula:
\( \frac{d}{dx}(x^x) = x^x (ln(x)+1) \)
Example 2
Find:
\( \frac{d}{dx}(x^x)|_{x=1} \)
Use:
\( \frac{d}{dx}(x^x) = x^x (ln(x)+1) \)
Substitute \(x=1\):
\( \frac{d}{dx}(x^x)|_{x=1} = 1^1 (ln(1)+1) \)
Since:
\(ln(1)=0\)
We get:
\( \frac{d}{dx}(x^x)|_{x=1} = 1^1 (ln(1)+1) \) = (0+1) = 1
So:
1
Example 3
Find:
\( \frac{d}{dx}(x^x)|_{x=e} \)
Use:
\( \frac{d}{dx}(x^x) = x^x (ln(x)+1) \)
Substitute \(x=e\):
\( \frac{d}{dx}(x^x)|_{x=e} = e^e (ln(e)+1) \)
Since:
\(ln(e)=1\)
We get:
\( \frac{d}{dx}(x^x)|_{x=e} = e^e (1+1) = e^e (2) = 2 e^e\)
So:
\(2e^e\)
Example 4
Find:
\( \frac{d}{dx}(2 x^x) \)
Use the constant multiple rule:
\( \frac{d}{dx}(2 x^x) = 2 \frac{d}{dx}(x^x) \)
Use:
\( \frac{d}{dx}(x^x) = x^x (ln(x)+1) \)
We get:
\( \frac{d}{dx}(2 x^x) = 2 \frac{d}{dx}(x^x) = 2x^x (ln(x)+1)\)
So:
\( 2x^x (ln(x)+1) \)
Practice
- More Examples (Current page)
- Take the Quiz