Q. 8 - Calculus

Question

When we say that $\frac{d}{dx}(\ln x)=\frac{1}{x\text{'}}$ we really mean to consider the function $\frac{1}{x}$ on the restricted domain $(0,\infty )$.Why?

Step-by-Step Solution

Answer

We use Restricted domains because the function $\frac{1}{x}$ has a one-to-one section

Step 1 . Given Information

We are given that $\frac{d}{dx} (\ln x) = \frac{1}{x}$ and $\frac{1}{x}$ is in. restricted domain $(0,\infty )$

Step 2. Solving

We have : $f\left(x\right)=\ln x$ so we can write -

$$\begin{gathered} f\text{'}\left(x\right)=\underset{h\to 0}{lim}\frac{\ln (x+h)-\ln x}{h} \\ =\underset{h\to 0}{lim}\frac{\ln \left({\displaystyle \frac{x+h}{x}}\right)}{h} \\ =\underset{h\to 0}{lim}\frac{1}{h}\ln (1+\frac{h}{x}) \\ =\underset{h\to 0}{lim}\ln {(1+\frac{h}{x})}^{\frac{1}{h}} \\ =\underset{h\to 0}{lim}\ln ({(1+\frac{1}{{\displaystyle \frac{x}{h}}})}^{\frac{x}{h}}{)}^{\frac{1}{x}} \end{gathered}$$

Now we put $\frac{x}{h}=N$, then as $h\to h,N\to \infty$

$$\begin{gathered} =\underset{N\to \infty }{lim}\ln ({(1+\frac{1}{N})}^N{)}^{\frac{1}{x}} \\ =\ln (\underset{N\to \infty }{lim}({(1+\frac{1}{N})}^N{)}^{\frac{1}{x}} \\ =\ln (e^{\frac{1}{x}}) (\sin ce \underset{N\to \infty }{lim}{(1+\frac{1}{N})}^N=e) \\ =\frac{1}{x} \end{gathered}$$

Therefore we get : $f\text{'}\left(x\right)=\frac{1}{x}$