Question

x^x+x^a+a^x+a^a , for some fixed a>0 and x>0

Answer 1

question is ----> Differentiate w.r.t. x the function x^x+ x^a + a^x+ a^a, for some fixed a > 0 and x > 0.


solution :- $\bf{y=x^x+x^a+a^x+a^a}$
we can write it ,
$\bf{y=e^{xlnx}+x^a+e^{xlna}+a^a}$
now differentiate with respect to x,
$\bf{\frac{dy}{dx} =\frac{d\{e^{xlnx}\}}{dx}+\frac{dx^a}{dx}+\frac{d\{e^{xlna}\}}{dx}+\frac{da^a}{dx}}\\\\=\bf{e^{xlnx}\frac{d\{xlnx\}}{dx} + ax^{a-1}+e^{alnx}\frac{d\{xlna\}}{dx}+0}\\\\=\bf{e^{xlnx}[x.\frac{1}{x}+1.lnx]+ax^{a-1}+e^{alnx}(lna)}\\\\=\bf{x^x[1+lnx]+ax^{a-1}+a^xlna}$

hence, $\bf{\frac{dy}{dx}=x^x(1+lnx)+ax^{a-1}+a^xlna}$