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Find the derivative of $\ln\left(\frac{x-1}{x}\right)$

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Final Answer

$\frac{x-x+1}{\left(x-1\right)x}$
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Step-by-step Solution

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The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{\frac{x-1}{x}}\frac{d}{dx}\left(\frac{x-1}{x}\right)$

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$\frac{1}{\frac{x-1}{x}}\frac{d}{dx}\left(\frac{x-1}{x}\right)$

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Learn how to solve differential calculus problems step by step online. Find the derivative of ln((x-1)/x). The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}. Divide fractions \frac{1}{\frac{x-1}{x}} with Keep, Change, Flip: a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}. Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}. Multiplying fractions \frac{x}{x-1} \times \frac{\frac{d}{dx}\left(x-1\right)x-\left(x-1\right)\frac{d}{dx}\left(x\right)}{x^2}.

Final Answer

$\frac{x-x+1}{\left(x-1\right)x}$

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Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of ln((x-1)/x) using the product ruleFind derivative of ln((x-1)/x) using the quotient ruleFind derivative of ln((x-1)/x) using logarithmic differentiationFind derivative of ln((x-1)/x) using the definition

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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

How to improve your answer:

Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

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