Given: An equation-
[tex]y=(10x+2)^x[/tex]Required: To determine the differentiation of y with respect to x.
Explanation: The differentiation of a logarithmic function is-
[tex]\begin{gathered} y=a^x \\ \frac{dy}{dx}=a^x\ln(a) \end{gathered}[/tex]Taking log both sides on the given equation as-
[tex]\begin{gathered} \ln y=\ln(10x+2)^x \\ =x\ln(10x+2) \end{gathered}[/tex]Now, differentiating with respect to x using product rule as-
[tex]\frac{1}{y}\frac{dy}{dx}=\ln(10x+2)\frac{d}{dx}(x)+x\frac{d}{dx}\ln(10x+2)[/tex]Further simplifying as-
[tex]\frac{dy}{dx}=y[\ln(10x+2)+\frac{10x}{10x+2}][/tex]Substituting the value of y as-
[tex]\frac{dy}{dx}=(10x+2)^x[\ln(10x+2)+\frac{10x}{10x+2}][/tex]Final Answer: Option D is correct.