Is this proof of the derivative of $\ln(x)$ correct?

Question

I just started with calculus at school. For myself I tried to proof why $\frac{dy}{dx}\ln(x) = \frac{1}{x}$ because I can't "just accept that it's true".

I came up with the following:

$$\ln(x) = y \Leftrightarrow e^y = x$$ $$\frac{dy}{dx}e^y = \frac{dy}{dx}x$$ $$\frac{dy}{dx}e^y = 1$$ $$\frac{dy}{dx} = \frac{1}{e^y}$$ $$\frac{dy}{dx} = \frac{1}{e^{\ln(x)}}$$ $$\frac{dy}{dx} = \frac{1}{x}$$

Would this be correct? Somehow I feel uncomfortable just moving $e^y$ to the other side "leaving $\frac{dy}{dx}$ behind" and was wondering if this is actually legal or if there are mistakes or improvements here. Thanks in advance!

Answer 1

$$\ln(x) = y \Leftrightarrow e^y = x$$ $$\frac{d(e^y)}{dx}\ = \frac{d(x)}{dx}$$ $$e^y\frac{dy}{dx} = 1$$ $$\frac{dy}{dx} = \frac{1}{e^y}$$ $$\frac{dy}{dx} = \frac{1}{e^{ln(x)}}$$ $$\frac{dy}{dx} = \frac{1}{x}$$

Answer 2

The notation is not appropriate.

The line of thought that you followed could be written this way.

From $y=\ln(x)$ you can write $e^{y(x)}=x$. Then take derivatives $$1=\frac{\partial x}{\partial x}=\frac{\partial(e^y)}{\partial x}=e^y\frac{\partial y}{\partial x}$$ where we have used the chain rule in the last equality, the given formula in the second, and the derivative of the identity function in the first.

Therefore $\frac{\partial y}{\partial x}=\frac{1}{e^y}=\frac{1}{x}$.

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The actual validity of the proof might depend, though, on the definition of $\ln(x)$ that you are given and the result that are allowed to be assumed known. Is the chain rule know to be true? Are $e^x$ and $\ln(x)$ known to be inverses?