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**hollywood** Don't let Leibniz's notation scare you. Whether you write $\displaystyle \frac{d}{dx}f(x)$ or f'(x), it's just the derivative of f.

Your professor introduced a new variable (or a new function, depending on how you look at it) $\displaystyle w=\ln{x}$. So $\displaystyle e^w=x$, and you take the derivative of both sides. The right side is easy. On the left, since w is a function of x, you have a composite function ($\displaystyle e^w$ is $\displaystyle e^{\ln{x}}$), so you need to use the chain rule. The derivative of the outside function is $\displaystyle \frac{d}{dw}e^w=e^w$, and the derivative of the inside function is $\displaystyle \frac{dw}{dx}$. So you have $\displaystyle e^w\frac{dw}{dx}=1$, so $\displaystyle \frac{dw}{dx}=\frac{1}{e^w}=\frac{1}{x}$, and since $\displaystyle w=\ln{x}$, $\displaystyle \frac{d}{dx}\ln{x}=\frac{1}{x}$, which is what you wanted to show.

If you can't figure out the derivation, you should at least memorize the result $\displaystyle \frac{d}{dx}\ln{x}=\frac{1}{x}$. Or in the other notation, if $\displaystyle f(x)=\ln{x}$, $\displaystyle f'(x)=\frac{1}{x}$.

Hope that helps.

- Hollywood