Final Answer
Step-by-step Solution
Problem to solve:
Find the derivative of $\ln\left(x\right)$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $\ln\left(x\right)$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$
Simplify the fraction
Using the power rule of logarithms: $n\log_b(a)=\log_b(a^n)$
Expand the fraction $\left(\frac{x+h}{x}\right)$ into $2$ simpler fractions with common denominator $x$
Simplify the fraction $\frac{x}{x}$ by $x$
Simplify
Apply the substitution $\frac{h}{x}=\frac{1}{n}$, then $h=\frac{x}{n}$. Since $h$ is approaching $0$, it is the same as if $n$ approaches $\infty$. Substituting
Rewrite the power $\left(1+\frac{1}{n}\right)^{\frac{n}{x}}$ by applying properties of exponents
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}$
The limit of a logarithm is equal to the logarithm of the limit
Using the representation of $e$ as a limit