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

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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

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

$\lim_{h\to0}\left(\frac{\ln\left(x+h\right)-\ln\left(x\right)}{h}\right)$
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: $n\log_b(a)=\log_b(a^n)$, where $n$ equals $\frac{1}{h}$

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

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

Calculate the power $e^1$

$\ln\left(e\right)\frac{1}{x}$
14

Calculating the natural logarithm of $e$

$\frac{1}{x}$

Final answer to the problem

$\frac{1}{x}$

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Function Plot

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

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1
2
3
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

How to improve your answer:

Main Topic: Definition of Derivative

Resolution of derivatives using the definition of the derivative, which is the limit of difference quotients of real numbers.

Used Formulas

See formulas (1)

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