** Final answer to the problem

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** Step-by-step Solution **

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- Find the derivative using the definition
- Find the derivative using the product rule
- 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
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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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

Learn how to solve quotient rule of differentiation problems step by step online. Find the derivative d/dx((x^2-1)/((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}}. Simplify \left(\left(x+1\right)^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. Simplify the product -(x^2-1). 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 to the problem

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