Computing the differential of a function

Question

I need to compute the differential of a function $y=e^x \ln x$.

Here's what I did: $$d\left( e^x \ln x \right) = \left(e^x \ln x + \frac{e^x}{ x} \right) dx.$$

Is that the correct answer?

Answer 1

What you did is correct.

Just as a suggestion, where you face expressions which only contain products, quotienst, powers, logarithmic differentitation uses to make life easier $$y=e^x \log (x) \implies \log(y)=x+\log(\log(x))$$ Differentiate both sides $$\frac{y'}y=1+\frac{1}{x \log (x)}$$ Now, use $$y'=y \times \left(\frac{y'}y \right)$$ replace and simplify to get the same result. As you see, I did not use the product rule at any time.