Find the derivative using the product rule $\frac{d}{dx}\left(\ln\left(\frac{\sqrt{2+x^2}}{x^2}\right)\right)$

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Final answer to the problem

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

How should I solve this problem?

  • Find the derivative using the product rule
  • Find the derivative using the definition
  • 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
  • Integrate by parts
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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 $\displaystyle f'(x)=\frac{a'}{a}$

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

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

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Learn how to solve product rule of differentiation problems step by step online. Find the derivative using the product rule d/dx(ln(((2+x^2)^(1/2))/(x^2))). 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}. Divide fractions \frac{1}{\frac{\sqrt{2+x^2}}{x^2}} 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}}. Simplify \left(x^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.

Final answer to the problem

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

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Function Plot

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

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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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