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

Step-by-step Solution

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Final answer to the problem

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

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1

Apply the formula: $derivdef\left(x\right)$$=\lim_{h\to0}\left(\frac{eval\left(x,var+h\right)-x}{h}\right)$, where $derivdefx=derivdef\left(\ln\left(x\right)\right)$ and $x=\ln\left(x\right)$

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

Apply the formula: $\ln\left(a\right)-\ln\left(b\right)$$=\ln\left(\frac{a}{b}\right)$, where $a=x+h$ and $b=x$

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

Apply the formula: $\frac{a}{b}$$=\frac{1}{b}a$, where $a=\ln\left(\frac{x+h}{x}\right)$ and $b=h$

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

Apply the formula: $a\ln\left(x\right)$$=\ln\left(x^a\right)$, where $a=\frac{1}{h}$ and $x=\frac{x+h}{x}$

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

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

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

Simplify the resulting fractions

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

Apply the formula: $\lim_{h\to0}\left(\ln\left(\left(1+\frac{h}{x}\right)^{\frac{1}{h}}\right)\right)$$=\lim_{n\to\infty }\left(\ln\left(\left(1+\frac{1}{n}\right)^{\frac{n}{x}}\right)\right)$, where $h/x=\frac{h}{x}$, $1+h/x=1+\frac{h}{x}$, $h->0=h\to0$ and $1/h=\frac{1}{h}$

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

Apply the formula: $a^{\frac{b}{c}}$$=\left(a^b\right)^{\frac{1}{c}}$, where $a=1+\frac{1}{n}$, $b=n$, $c=x$ and $b/c=\frac{n}{x}$

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

Apply the formula: $\ln\left(x^a\right)$$=a\ln\left(x\right)$, where $a=\frac{1}{x}$ and $x=\left(1+\frac{1}{n}\right)^n$

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

Apply the formula: $\lim_{x\to c}\left(ab\right)$$=a\lim_{x\to c}\left(b\right)$, where $a=\frac{1}{x}$, $b=\ln\left(\left(1+\frac{1}{n}\right)^n\right)$, $c=\infty $ and $x=n$

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

Apply the formula: $\lim_{x\to c}\left(\ln\left(a\right)\right)$$=\ln\left(\lim_{x\to c}\left(a\right)\right)$, where $a=\left(1+\frac{1}{n}\right)^n$, $c=\infty $ and $x=n$

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

Apply the formula: $\lim_{x\to\infty }\left(\left(1+\frac{a}{x}\right)^x\right)$$=e^a$, where $a=1$ and $x=n$

$\ln\left(e^1\right)\frac{1}{x}$
13

Apply the formula: $\ln\left(x\right)$$=logf\left(x,e\right)$, where $x=e^1$

$\frac{1}{x}$

Final answer to the problem

$\frac{1}{x}$

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Function Plot

Plotting: $\frac{1}{x}$

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1
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4
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

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