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

## Step-by-step Solution

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###  Videos

$\frac{1}{x^{1}\left(x-1\right)}$
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##  Step-by-step Solution 

Problem to solve:

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

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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 $\displaystyle f'(x)=\frac{a'}{a}$

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

Learn how to solve differential calculus problems step by step online.

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

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}. 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}}. Simplify the product -(x-1). The derivative of the linear function is equal to 1.

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

##  Explore different ways to solve this problem

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 d/dx(ln((x-1)/x)) using the product ruleFind d/dx(ln((x-1)/x)) using the quotient ruleFind d/dx(ln((x-1)/x)) using logarithmic differentiationFind d/dx(ln((x-1)/x)) using the definition

SnapXam A2

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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

### Main topic:

Differential Calculus

~ 0.07 s

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