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

Step-by-step Solution

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Solution

Final Answer

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

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

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

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

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

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

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

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

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

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

$\frac{2\left(1-x^2\right)\left(\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(-x^2\right)\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-\left(\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(1\right)\right)\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
6

Simplify the product $-(\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(1\right))$

$\frac{2\left(1-x^2\right)\left(\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(-x^2\right)\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}\left(2x\right)-\frac{d}{dx}\left(1\right)\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
7

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

$\frac{2\left(1-x^2\right)\frac{d}{dx}\left(-x^2\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}\left(2x\right)-\frac{d}{dx}\left(1\right)\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
8

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

$\frac{2\left(1-x^2\right)\frac{d}{dx}\left(-x^2\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}\left(2x\right)- 0\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
9

Multiply $-1$ times $0$

$\frac{2\left(1-x^2\right)\frac{d}{dx}\left(-x^2\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}\left(2x\right)\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
10

The derivative of the linear function times a constant, is equal to the constant

$\frac{2\left(1-x^2\right)\frac{d}{dx}\left(-x^2\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-2\frac{d}{dx}\left(x\right)\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
11

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

$\frac{2\left(1-x^2\right)\frac{d}{dx}\left(-x^2\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-2\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
12

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

$\frac{-2\left(1-x^2\right)\frac{d}{dx}\left(x^2\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-2\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
13

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

$\frac{-4\left(1-x^2\right)\left(x^2+2x+1\right)x+\left(-2x-2\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
14

Simplify the derivative

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

Final Answer

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

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 derivative of ((1+-1x^2)^2)/(x^2+2x) using the product ruleFind derivative of ((1+-1x^2)^2)/(x^2+2x) using the quotient ruleFind derivative of ((1+-1x^2)^2)/(x^2+2x) using logarithmic differentiationFind derivative of ((1+-1x^2)^2)/(x^2+2x) using the definition

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

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÷
◻2
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√

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∞
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log◻
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d/dx
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∫◻
|◻|
θ
=
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<
>=
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sin
cos
tan
cot
sec
csc

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acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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