Question

If f(x) = x (ln x), find f'(x). A. e^(x + 1) B. 1 + ln x C. 1 + e^x D. (ln x)^2 + x

Answer 1

Answer:

$f'(x)=1+\ln x$

Step-by-step explanation:

$f'(x)$ is notation for the first derivative of $f(x)$.

Recall the product rule:

$(f\cdot g)'=f'\cdot g+g'\cdot f$

Therefore, we have:

$\displaystyle \frac{d}{dx}(x\ln x)=\frac{d}{dx}(x)\cdot \ln(x)+\frac{d}{dx}(\ln x)\cdot x$

Note that:

$\displaystyle \frac{d}{dx}(x)=1,\\\\\frac{d}{dx}(\ln x)=\frac{1}{x}$

Simplify:

$\displaystyle \frac{d}{dx}(x\ln x)=1\cdot \ln x+\frac{1}{x}\cdot x, \\\\\frac{d}{dx}(x\ln x)= \ln x+1=\boxed{1+\ln x}$