what did i do wrong when trying to prove the derivative of ln(x)

Question

when trying to prove the derivative i ended up with

$$\lim_{h \to 0}\frac{\ln(x+h)-\ln(x)}{h}=\lim_{h \to 0}\frac{\ln(\frac{x+h}{h})}{h}=\lim_{h \to 0}\ln((1+\frac{h}{x})^{(\frac{1}{h})}$$ and $$\frac{h}{x}\approx0$$ so

as you can see i attempted to solve it by treating $\frac{h}{x}$ as if its zero due to it being arbitrarily small but i got the wrong answer. now when trying to prove the derivative of $x^2$ i ended up with something similar:

$$\lim_{h \to 0}\frac{(x+h)^2-x^2}{h}=\lim_{h \to 0}\frac{x^2+h^2+2xh-x^2}{h}=\lim_{h \to 0}\frac{h^2+2xh}{h}=\lim_{h \to 0}h+2x=2x$$

here i treated h as if it were zero due to it being arbitrarily small and i got the right answer. was i wrong for treating $\frac{h}{x}$ as if its zero? (please forgive me for not writing the equations formally i am fairly new at this edit: when i was solving

Answer 1

$$\lim\limits_{h \to 0} \frac{\ln(x+h)-\ln(x)}{h} = \lim\limits_{h \to 0}\frac{1}{h} \ln\left(1+\frac{h}{x}\right) = \lim\limits_{h \to 0} \ln\left(1+\frac{h}{x} \right)^{\frac{1}{h}}$$

We use the limit formula: $$\lim\limits_{h \to 0}(1+a)^{\frac{1}{h}} = e^{\lim\limits_{h \to 0}\frac{a}{h}}$$

So our case we have $a =\frac{h}{x}$. Therefore let

$$\lim\limits_{h \to 0}\ln\left(1+\frac{h}{x} \right)^{\frac{1}{h}} = y$$ $$ \Rightarrow \lim\limits_{h \to 0}\left(1+\frac{h}{x} \right)^{\frac{1}{h}} = e^y \Rightarrow e^{\frac{1}{x}} = e^y \Rightarrow y =\frac{1}{x}$$

As required.

Answer 2

The first thing you missed was that $h$ was in the denominator such that you will get the power $\frac{1}{h}$ in the logarithm (which was fixed in the first line by an edit now and still is not fixed later on). Afterwards you let $h/x \rightarrow 0$ without considerung the $h$ in the power. You need to consider all the $h$'s at the same time when computing the limit. Here is how you can do it:

We have $$\lim_{h \rightarrow 0}\frac{\text{log}(x+h) - \text{log}(x)}{h} = \lim_{h \rightarrow 0}\text{log}\left(\left(1 + \frac{h}{x}\right)^{\frac{1}{h}}\right) = \lim_{h \rightarrow \infty}\text{log}\left(\left(1 + \frac{\frac{1}{x}}{h}\right)^{h}\right),$$ which is basically what you computed and using the continuity of the logarithm and the limit representation of the exponential function we now get $$\lim_{h \rightarrow \infty}\text{log}\left(\left(1 + \frac{\frac{1}{x}}{h}\right)^{h}\right)=\text{log}\left(\lim_{h \rightarrow \infty}\left(1 + \frac{\frac{1}{x}}{h}\right)^{h}\right) = \text{log}\left(\text{exp}\left(\frac{1}{x}\right)\right) = \frac{1}{x}.$$ Thus $\frac{d}{dx}\text{log}(x) = \frac{1}{x}$.

Answer 3

$\lim\limits_{h\to0}\dfrac{\ln(x+h)-\ln x}{h}=\lim\limits_{h\to0}\dfrac{\ln\left(1+\dfrac hx\right)}h=\lim\limits_{h\to0}\dfrac{\dfrac hx+o(h)}h=\dfrac 1x$