Final answer to the problem
$\frac{-4\left(1-x^2\right)\left(x^2+2x+1\right)x+\left(-2x-2\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
Got another answer? Verify it here!
Step-by-step Solution
Specify the solving method
Find the derivative Find the derivative using the product rule Find the derivative using the quotient rule Logarithmic Differentiation Find the derivative using the definition Suggest another method or feature
Send
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 $0$
$\frac{2\left(1-x^2\right)\frac{d}{dx}\left(-x^2\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}\left(2x\right)\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
Intermediate steps
10
The derivative of the linear function times a constant, is equal to the constant
$\frac{2\left(1-x^2\right)\frac{d}{dx}\left(-x^2\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-2\frac{d}{dx}\left(x\right)\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
Explain this step further
Intermediate steps
11
The derivative of the linear function is equal to $1$
$\frac{2\left(1-x^2\right)\frac{d}{dx}\left(-x^2\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-2\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
Explain this step further
Intermediate steps
12
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
$\frac{-2\left(1-x^2\right)\frac{d}{dx}\left(x^2\right)\left(x^2+2x+1\right)+\left(-\frac{d}{dx}\left(x^2\right)-2\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
Explain this step further
Intermediate steps
13
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{-4\left(1-x^2\right)\left(x^2+2x+1\right)x+\left(-2x-2\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$
Explain this step further
Final answer to the problem
$\frac{-4\left(1-x^2\right)\left(x^2+2x+1\right)x+\left(-2x-2\right)\left(1-x^2\right)^2}{\left(x^2+2x+1\right)^2}$