Question

The value of $\frac{d}{dx} [x^n \log_a xe^x ] = $

• A. $e^x \log_a x + \frac{x^{n-1}}{\log_e a}$
• B. $ e^{x} x^{n-1} \left[x \log_{a} x + \frac{1}{\log_{e}a} + x \log_{a}x \right]$
• C. $nx^{n-1} \log_{a} \left(xe^{x}\right) $
• D. $x^{n } \log_{a} \left(xe^{x}\right)$

Answer 1

The Correct Option is B

$\frac{d}{dx} \left[x^{n} \left(\log_{a} x\right)e^{x}\right] $ $=x^{n} \log_{a} x. e^{x} +x^{n}e^{x}. \frac{1}{x} \log_{a}e + nx^{n-1} \log_{a}x . e^{x} $ $= e^{x} \left[x^{n} \log_{a} x+x^{n-1} \log_{a} e + nx^{n-1} \log_{a}x\right] $ $= x^{ n-1} e^{x} \left[x \log_{a} x + \frac{1}{\log_{e}a} + n \log_{a}x\right]$