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Find the derivative $\frac{d}{dx}\left(\frac{x}{\ln\left(x\right)^2}\right)$

Step-by-step Solution

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Final answer to the problem

$\frac{\ln\left(x\right)-2}{\ln\left(x\right)^{3}}$
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Step-by-step Solution

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  • Find the derivative using the definition
  • Find the derivative using the product rule
  • Find the derivative using the quotient rule
  • Find the derivative using logarithmic differentiation
  • Find the derivative
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
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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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\frac{d}{dx}\left(x\right)\ln\left(x\right)^2-x\frac{d}{dx}\left(\ln\left(x\right)^2\right)}{\left(\ln\left(x\right)^2\right)^2}$

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$\frac{\frac{d}{dx}\left(x\right)\ln\left(x\right)^2-x\frac{d}{dx}\left(\ln\left(x\right)^2\right)}{\left(\ln\left(x\right)^2\right)^2}$

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Learn how to solve quotient rule of differentiation problems step by step online. Find the derivative d/dx(x/(ln(x)^2)). 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 \left(\ln\left(x\right)^2\right)^2 using the power of a power property: \left(a^m\right)^n=a^{m\cdot n}. In the expression, m equals 2 and n equals 2. The derivative of the linear function is equal to 1. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}.

Final answer to the problem

$\frac{\ln\left(x\right)-2}{\ln\left(x\right)^{3}}$

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Function Plot

Plotting: $\frac{\ln\left(x\right)-2}{\ln\left(x\right)^{3}}$

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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: Quotient Rule of Differentiation

The quotient rule is a formal rule for differentiating problems where one function is divided by another.

Used Formulas

See formulas (3)

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