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Find the derivative of $\frac{3}{x^2\left(x^2+2x+3\right)+\frac{-1}{x^2+2x+3}}$

Step-by-step Solution

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Final answer to the problem

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

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Combine $x^2\left(x^2+2x+3\right)+\frac{-1}{x^2+2x+3}$ in a single fraction

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

Learn how to solve limits by direct substitution problems step by step online.

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

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Learn how to solve limits by direct substitution problems step by step online. Find the derivative of 3/(x^2(x^2+2x+3)+-1/(x^2+2x+3)). Combine x^2\left(x^2+2x+3\right)+\frac{-1}{x^2+2x+3} in a single fraction. Divide fractions \frac{3}{\frac{-1+\left(x^2+2x+3\right)^2x^2}{x^2+2x+3}} 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}}. The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function.

Final answer to the problem

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

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Function Plot

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

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7
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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

How to improve your answer:

Main Topic: Limits by Direct Substitution

Find limits of functions at a specific point by directly plugging the value into the function.

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