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Calculus Level pending

Let $f(x)=x^{x}$ and x is not equal to zero. Find the derivative of this function and find the value of the derivative at $x=e$

23 32.1 30.3 29.8

1 solution

Caleb Townsend
Mar 14, 2015

$f(x) =x^x = e^{x\ln(x)} \\ \frac{d}{dx}f(x) = (e^{x\ln(x)})(\frac{d}{dx}x\ln(x)) \\ = (x^x)(\ln(x) + 1)$

$f'(e) = e^e\times 2 \\ \approx \boxed{30.3}$

Try doing the same for $sin^x(x)$

Anirudh Chandramouli - 6 years, 3 months ago

$f(x) = \sin^x (x) = e^{x\ln(\sin(x))} \\ \frac{d}{dx}f(x) = (e^{x\ln(\sin(x))})\frac{d}{dx}(x\ln(\sin(x))) \\ = \sin^x(x)(\ln(\sin(x)) + x\cot(x))$

Caleb Townsend - 6 years, 3 months ago