Step-by-step Solution

Find the derivative of $\ln\left(x\right)$ using the definition

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Final Answer

$\frac{1}{x}$
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Step-by-step Solution

Problem to solve:

$derivdef\left(\ln\left(x\right)\right)$
1

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

$\lim_{h\to0}\left(\frac{\ln\left(x+h\right)-\ln\left(x\right)}{h}\right)$
2

The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$

$\lim_{h\to0}\left(\frac{\ln\left(\frac{x+h}{x}\right)}{h}\right)$
3

Simplify the fraction

$\lim_{h\to0}\left(\frac{1}{h}\ln\left(\frac{x+h}{x}\right)\right)$
4

Using the power rule of logarithms: $n\log_b(a)=\log_b(a^n)$

$\lim_{h\to0}\left(\ln\left(\left(\frac{x+h}{x}\right)^{\frac{1}{h}}\right)\right)$
5

Expand the fraction $\left(\frac{x+h}{x}\right)$ into $2$ simpler fractions with common denominator $x$

$\lim_{h\to0}\left(\ln\left(\left(\frac{x}{x}+\frac{h}{x}\right)^{\frac{1}{h}}\right)\right)$

Simplify the fraction $\frac{x}{x}$ by $x$

$\lim_{h\to0}\left(\ln\left(\left(1+\frac{h}{x}\right)^{\frac{1}{h}}\right)\right)$
6

Simplify

$\lim_{h\to0}\left(\ln\left(\left(1+\frac{h}{x}\right)^{\frac{1}{h}}\right)\right)$
7

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

$\lim_{n\to\infty }\left(\ln\left(\left(1+\frac{1}{n}\right)^{\frac{n}{x}}\right)\right)$
8

Rewrite the power $\left(1+\frac{1}{n}\right)^{\frac{n}{x}}$ by applying properties of exponents

$\lim_{n\to\infty }\left(\ln\left(\left(\left(1+\frac{1}{n}\right)^n\right)^{\frac{1}{x}}\right)\right)$
9

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$\lim_{n\to\infty }\left(\frac{1}{x}\ln\left(\left(1+\frac{1}{n}\right)^n\right)\right)$
10

The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}$

$\frac{1}{x}\lim_{n\to\infty }\left(\ln\left(\left(1+\frac{1}{n}\right)^n\right)\right)$
11

The limit of a logarithm is equal to the logarithm of the limit

$\frac{1}{x}\ln\left(\lim_{n\to\infty }\left(\left(1+\frac{1}{n}\right)^n\right)\right)$
12

Using the representation of $e$ as a limit

$\frac{1}{x}$

Final Answer

$\frac{1}{x}$
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1
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5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Tips on how to improve your answer:

$derivdef\left(\ln\left(x\right)\right)$

Time to solve it:

~ 0.33 s