​$$\text { Find the first derivative of } e^{x \ln a}+e^{a \ln x}+e^{a \ln a} \text {. }$$​​

Correct option is C

​$$\begin{aligned}&{\textbf{Term 1: } e^{x \ln a}} \\[5pt]&\text{Using the identity: } e^{x \ln a} = a^x \\[5pt]&\text{So: } \frac{d}{dx} \left( e^{x \ln a} \right) = \frac{d}{dx} (a^x) = a^x \ln a \\[10pt]&{ \textbf{Term 2: } e^{a \ln x}} \\[5pt]&\text{Using the identity: } e^{a \ln x} = x^a \\[5pt]&\text{So: } \frac{d}{dx} \left( e^{a \ln x} \right) = \frac{d}{dx} (x^a) = ax^{a-1} \\[10pt]&{ \textbf{Term 3: } e^{a \ln a}} \\[5pt]&\text{This is constant: } e^{a \ln a} = a^a \\[5pt]&\text{So derivative is: } \frac{d}{dx}(a^a) = 0 \\[10pt]&{ \textbf{Final Answer:}} \\&\quad f'(x) = a^x \ln a + ax^{a-1}\end{aligned}$$​​