We have to find the values of constants A and B.
Given the equality in the question, we can write:
[tex]8xe^{6x}=\frac{d}{dx}[(Ax+B)e^{6x}][/tex]
We then can calculate the derivative as:
[tex]\begin{gathered} \frac{d}{dx}[(Ax+B)e^{6x}] \\ A\cdot\frac{d}{dx}(xe^{6x})+B\cdot\frac{d}{dx}(e^{6x}) \\ A(x\cdot6e^{6x}+e^{6x})+B(6e^{6x}) \\ (6Ax+A+6B)e^{6x} \end{gathered}[/tex]
Then, we can compare the terms:
[tex]\begin{gathered} (8x)e^{6x}=(6Ax+A+6B)e^{6x} \\ \Rightarrow8=6A \\ \Rightarrow0=A+6B \end{gathered}[/tex]
We compared the linear and independent terms to find equations to solve for A and B.
Then, we can find A as:
[tex]8=6A\Rightarrow A=\frac{8}{6}=\frac{4}{3}[/tex]
and B as:
[tex]\begin{gathered} 0=A+6B \\ 6B=-A \\ 6B=-\frac{4}{3} \\ B=-\frac{4}{6*3} \\ B=-\frac{4}{18} \\ B=-\frac{2}{9} \end{gathered}[/tex]
Answer: A = 4/3 and B = -2/9.
