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Find the derivative using logarithmic differentiation method $\frac{x^5-x^4+x^2-2}{x^2+1}$

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Final answer to the problem

$\frac{3x^{6}-2x^{5}-4x^{3}+5x^{4}+6x}{\left(x^2+1\right)^2}$
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Step-by-step Solution

How should I solve this problem?

  • Find the derivative using logarithmic differentiation
  • Find the derivative using the definition
  • Find the derivative using the product rule
  • Find the derivative using the quotient rule
  • Find the derivative
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
  • Load more...
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1

To derive the function $\frac{x^5-x^4+x^2-2}{x^2+1}$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=\frac{x^5-x^4+x^2-2}{x^2+1}$
2

Apply natural logarithm to both sides of the equality

$\ln\left(y\right)=\ln\left(\frac{x^5-x^4+x^2-2}{x^2+1}\right)$
3

Apply logarithm properties to both sides of the equality

$\ln\left(y\right)=\ln\left(x^5-x^4+x^2-2\right)-\ln\left(x^2+1\right)$
4

Derive both sides of the equality with respect to $x$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(\ln\left(x^5-x^4+x^2-2\right)-\ln\left(x^2+1\right)\right)$
5

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(\ln\left(x^5-x^4+x^2-2\right)-\ln\left(x^2+1\right)\right)$
6

The derivative of the linear function is equal to $1$

$\frac{y^{\prime}}{y}=\frac{d}{dx}\left(\ln\left(x^5-x^4+x^2-2\right)-\ln\left(x^2+1\right)\right)$
7

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{y^{\prime}}{y}=\frac{d}{dx}\left(\ln\left(x^5-x^4+x^2-2\right)\right)+\frac{d}{dx}\left(-\ln\left(x^2+1\right)\right)$
8

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{y^{\prime}}{y}=\frac{d}{dx}\left(\ln\left(x^5-x^4+x^2-2\right)\right)-\frac{d}{dx}\left(\ln\left(x^2+1\right)\right)$
9

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{y^{\prime}}{y}=\frac{1}{x^5-x^4+x^2-2}\frac{d}{dx}\left(x^5-x^4+x^2-2\right)-\left(\frac{1}{x^2+1}\right)\frac{d}{dx}\left(x^2+1\right)$
10

Multiplying the fraction by $-1$

$\frac{y^{\prime}}{y}=\frac{1}{x^5-x^4+x^2-2}\frac{d}{dx}\left(x^5-x^4+x^2-2\right)+\frac{-1}{x^2+1}\frac{d}{dx}\left(x^2+1\right)$
11

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{y^{\prime}}{y}=\frac{1}{x^5-x^4+x^2-2}\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-2\right)\right)+\frac{-1}{x^2+1}\frac{d}{dx}\left(x^2+1\right)$
12

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{y^{\prime}}{y}=\frac{1}{x^5-x^4+x^2-2}\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-2\right)\right)+\frac{-1}{x^2+1}\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)\right)$
13

The derivative of the constant function ($-2$) is equal to zero

$\frac{y^{\prime}}{y}=\frac{1}{x^5-x^4+x^2-2}\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)\right)+\frac{-1}{x^2+1}\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)\right)$
14

The derivative of the constant function ($1$) is equal to zero

$\frac{y^{\prime}}{y}=\frac{1}{x^5-x^4+x^2-2}\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)\right)+\frac{-1}{x^2+1}\frac{d}{dx}\left(x^2\right)$
15

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{y^{\prime}}{y}=\frac{1}{x^5-x^4+x^2-2}\left(\frac{d}{dx}\left(x^5\right)-\frac{d}{dx}\left(x^4\right)+\frac{d}{dx}\left(x^2\right)\right)+\frac{-1}{x^2+1}\frac{d}{dx}\left(x^2\right)$
16

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{y^{\prime}}{y}=\frac{1}{x^5-x^4+x^2-2}\left(5x^{4}-4x^{3}+2x\right)+2\left(\frac{-1}{x^2+1}\right)x$
17

Multiply the fraction and term

$\frac{y^{\prime}}{y}=\frac{5x^{4}-4x^{3}+2x}{x^5-x^4+x^2-2}+2\left(\frac{-1}{x^2+1}\right)x$
18

Multiplying the fraction by $2x$

$\frac{y^{\prime}}{y}=\frac{5x^{4}-4x^{3}+2x}{x^5-x^4+x^2-2}+\frac{-2x}{x^2+1}$
19

Multiply both sides of the equation by $y$

$y^{\prime}=\left(\frac{5x^{4}-4x^{3}+2x}{x^5-x^4+x^2-2}+\frac{-2x}{x^2+1}\right)y$
20

Substitute $y$ for the original function: $\frac{x^5-x^4+x^2-2}{x^2+1}$

$y^{\prime}=\left(\frac{5x^{4}-4x^{3}+2x}{x^5-x^4+x^2-2}+\frac{-2x}{x^2+1}\right)\frac{x^5-x^4+x^2-2}{x^2+1}$
21

The derivative of the function results in

$\left(\frac{5x^{4}-4x^{3}+2x}{x^5-x^4+x^2-2}+\frac{-2x}{x^2+1}\right)\frac{x^5-x^4+x^2-2}{x^2+1}$
22

Simplify the derivative

$\frac{3x^{6}-2x^{5}-4x^{3}+5x^{4}+6x}{\left(x^2+1\right)^2}$

Final answer to the problem

$\frac{3x^{6}-2x^{5}-4x^{3}+5x^{4}+6x}{\left(x^2+1\right)^2}$

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Function Plot

Plotting: $\frac{3x^{6}-2x^{5}-4x^{3}+5x^{4}+6x}{\left(x^2+1\right)^2}$

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a
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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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