Find the derivative $\frac{d}{dx}\left(\frac{\frac{1}{x^2}\ln\left(x+1\right)}{\ln\left(x\right)^2}\right)$

Step-by-step Solution

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Final answer to the problem

$\frac{\frac{x^2\ln\left(x\right)^2}{x+1}+\left(-2\ln\left(x\right)^2x-2x\ln\left(x\right)\right)\ln\left(x+1\right)}{x^{4}\ln\left(x\right)^{4}}$
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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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1

Multiply the fraction by the term $\ln\left(x+1\right)$

$\frac{d}{dx}\left(\frac{\frac{\ln\left(x+1\right)}{x^2}}{\ln\left(x\right)^2}\right)$

Learn how to solve quotient rule of differentiation problems step by step online.

$\frac{d}{dx}\left(\frac{\frac{\ln\left(x+1\right)}{x^2}}{\ln\left(x\right)^2}\right)$

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Learn how to solve quotient rule of differentiation problems step by step online. Find the derivative d/dx((1/(x^2)ln(x+1))/(ln(x)^2)). Multiply the fraction by the term \ln\left(x+1\right). Divide fractions \frac{\frac{\ln\left(x+1\right)}{x^2}}{\ln\left(x\right)^2} with Keep, Change, Flip: \frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}. 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}}. The power of a product is equal to the product of it's factors raised to the same power.

Final answer to the problem

$\frac{\frac{x^2\ln\left(x\right)^2}{x+1}+\left(-2\ln\left(x\right)^2x-2x\ln\left(x\right)\right)\ln\left(x+1\right)}{x^{4}\ln\left(x\right)^{4}}$

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Function Plot

Plotting: $\frac{\frac{x^2\ln\left(x\right)^2}{x+1}+\left(-2\ln\left(x\right)^2x-2x\ln\left(x\right)\right)\ln\left(x+1\right)}{x^{4}\ln\left(x\right)^{4}}$

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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

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