Limits, Continuity and Differentiability
80254
For constant \(a, \frac{d}{d x}\left(x^{x}+x^{a}+a^{x}+a^{a}\right)\) is
1 \(x^{x}(1+\log x)+a x^{a-1}\)
2 \(x^{x}(1+\log x)+a x^{a-1}+a^{x} \log a\)
3 \(x^{x}(1+\log x)+a^{a}(1+\log x)\)
4 \(x^{x}(1+\log x)+a^{a}(1+\log a)+a^{a-1}\)
Explanation:
(B) : Given,
\(\frac{d}{d x}\left(\mathrm{x}^{\mathrm{x}}+\mathrm{x}^{\mathrm{a}}+\mathrm{a}^{\mathrm{x}}+\mathrm{a}^{\mathrm{a}}\right)\)
Let, \(\quad y=x^{x}\)
Taking \(\log\) both sides.
\(\log \mathrm{y}=\mathrm{x} \log \mathrm{x}\)
Differentiating both sides w.r.t. \(\mathrm{x}\)
\(\frac{1}{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x} \frac{1}{\mathrm{x}}+\log \mathrm{x} .1\)
\(\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}[1+\log \mathrm{x}]\)
\(\frac{d y}{d x}=x^{x}[1+\log x]\)
Then,
\(\frac{d}{d x}\left(x^{x}+x^{a}+a^{x}+a^{a}\right)=x^{x}[1+\log x]+a x^{a-1}+a^{x} \log a+0\)
\(=x^{x}[1+\log x]+a x^{a-1}+a^{x} \log a\)