Question

Find the derivative of g(x) = (1/x) at x=3 algebraically.

Answer 1

Algebraically? Do you mean using the limit definition?

Answer 2

i think so

Answer 3

\[\begin{align*}\frac{d}{dx}\ln x&=\lim_{h\to0}\frac{\ln(x+h)-\ln x}{h}\\
&=\lim_{h\to0}\frac{\ln\left(\frac{x+h}{x}\right)}{h}\\
&=\lim_{h\to0}\frac{1}{h}\ln\left(1+\frac{h}{x}\right)
\end{align*}\]
Substitute \(t=\dfrac{h}{x}\). Then, as \(h\to0\), you have \(t\to0\). This gives you \(h=xt\), or \(\dfrac{1}{h}=\dfrac{1}{xt}\).
\[\begin{align*}\frac{d}{dx}\ln x&=\lim_{t\to0}\frac{1}{xt}\ln\left(1+t\right)\\
&=\frac{1}{x}\color{red}{\lim_{t\to0}\ln(1+t)^{1/t}}
\end{align*}\]
The red part is the limit definition of \(\ln e\), or just 1. So you're left with \(\dfrac{1}{x}\), the derivative of \(\ln x\).