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# Derive the function (2-1x)^3(1-1x^2)^2 with respect to x

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## Answer

$-3\left(1-x^2\right)^2\left(2-x\right)^{2}-4x\left(2-x\right)^3\left(1-x^2\right)$

## Step by step solution

Problem

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\left(2-x\right)^3$ and $g=\left(1-x^2\right)^2$

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

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

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

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

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

The derivative of the constant function is equal to zero

$2\left(2-x\right)^3\left(1-x^2\right)^{\left(2-1\right)}\left(\frac{d}{dx}\left(-x^2\right)+0\right)+3\left(1-x^2\right)^2\left(2-x\right)^{2}\left(\frac{d}{dx}\left(-x\right)+0\right)$
5

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

$2\left(2-x\right)^3\left(1-x^2\right)^{\left(2-1\right)}\left(0-\frac{d}{dx}\left(x^2\right)\right)+3\left(1-x^2\right)^2\left(2-x\right)^{2}\left(0-\frac{d}{dx}\left(x\right)\right)$
6

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

$2\left(2-x\right)^3\left(1-x^2\right)^{\left(2-1\right)}\left(0-\frac{d}{dx}\left(x^2\right)\right)+\left(1\left(-1\right)+0\right)\cdot 3\left(1-x^2\right)^2\left(2-x\right)^{2}$
7

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

$2\left(2-x\right)^3\left(0-1\cdot 2x\right)\left(1-x^2\right)^{\left(2-1\right)}+\left(1\left(-1\right)+0\right)\cdot 3\left(1-x^2\right)^2\left(2-x\right)^{2}$
8

Subtract the values $2$ and $-1$

$2\left(2-x\right)^3\left(0-1\cdot 2x\right)\left(1-x^2\right)^{1}+\left(1\left(-1\right)+0\right)\cdot 3\left(1-x^2\right)^2\left(2-x\right)^{2}$
9

Multiply $-1$ times $1$

$2\left(2-x\right)^3\left(0-2x\right)\left(1-x^2\right)^{1}+\left(0-1\right)\cdot 3\left(1-x^2\right)^2\left(2-x\right)^{2}$
10

Subtract the values $0$ and $-1$

$2\left(2-x\right)^3\left(0-2x\right)\left(1-x^2\right)^{1}-1\cdot 3\left(1-x^2\right)^2\left(2-x\right)^{2}$
11

Multiply $3$ times $-1$

$2\left(2-x\right)^3\left(0-2x\right)\left(1-x^2\right)^{1}-3\left(1-x^2\right)^2\left(2-x\right)^{2}$
12

$x+0=x$, where $x$ is any expression

$2\left(-2\right)x\left(2-x\right)^3\left(1-x^2\right)^{1}-3\left(1-x^2\right)^2\left(2-x\right)^{2}$
13

Multiply $-2$ times $2$

$-3\left(1-x^2\right)^2\left(2-x\right)^{2}-4x\left(2-x\right)^3\left(1-x^2\right)^{1}$
14

Any expression to the power of $1$ is equal to that same expression

$-3\left(1-x^2\right)^2\left(2-x\right)^{2}-4x\left(2-x\right)^3\left(1-x^2\right)$

## Answer

$-3\left(1-x^2\right)^2\left(2-x\right)^{2}-4x\left(2-x\right)^3\left(1-x^2\right)$

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### Main topic:

Differential calculus

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