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Find the derivative $\frac{d}{dx}\left(\frac{x^2\sqrt{3x-2}}{\left(x-1\right)^2}\right)$

Step-by-step Solution

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Final Answer

$\frac{\left(2x\sqrt{3x-2}+\frac{\frac{3}{2}x^2}{\sqrt{3x-2}}\right)\left(x-1\right)^2-2x^2\sqrt{3x-2}\left(x-1\right)}{\left(x-1\right)^{4}}$
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Step-by-step Solution

Problem to solve:

$\frac{d}{dx}\left(\frac{x^2\sqrt{3x-2}}{\left(x-1\right)^2}\right)$

Specify the solving method

1

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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\frac{d}{dx}\left(x^2\sqrt{3x-2}\right)\left(x-1\right)^2-x^2\sqrt{3x-2}\frac{d}{dx}\left(\left(x-1\right)^2\right)}{\left(x-1\right)^{4}}$
2

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x^2$ and $g=\sqrt{3x-2}$

$\frac{\left(\frac{d}{dx}\left(x^2\right)\sqrt{3x-2}+x^2\frac{d}{dx}\left(\sqrt{3x-2}\right)\right)\left(x-1\right)^2-x^2\sqrt{3x-2}\frac{d}{dx}\left(\left(x-1\right)^2\right)}{\left(x-1\right)^{4}}$

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

$\frac{\frac{d}{dx}\left(x^2\sqrt{3x-2}\right)\left(x-1\right)^2-x^2\sqrt{3x-2}\frac{d}{dx}\left(\left(x-1\right)^2\right)}{\left(x-1\right)^{4}}$

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Learn how to solve quotient rule of differentiation problems step by step online. Find the derivative d/dx((x^2(3x-2)^1/2)/((x-1)^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}}. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x^2 and g=\sqrt{3x-2}. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-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

$\frac{\left(2x\sqrt{3x-2}+\frac{\frac{3}{2}x^2}{\sqrt{3x-2}}\right)\left(x-1\right)^2-2x^2\sqrt{3x-2}\left(x-1\right)}{\left(x-1\right)^{4}}$

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((x^2(3x-2)^1/2)/((x-1)^2)) using the product ruleFind d/dx((x^2(3x-2)^1/2)/((x-1)^2)) using the quotient ruleFind d/dx((x^2(3x-2)^1/2)/((x-1)^2)) using logarithmic differentiationFind d/dx((x^2(3x-2)^1/2)/((x-1)^2)) 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

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