Calculus Examples

{(x - 1)}^{2}

${(x - 1)}^{2}$

Step 1

Rewrite

{(x - 1)}^{2}

${(x - 1)}^{2}$ as

(x - 1) (x - 1)

$(x - 1) (x - 1)$ .

\frac{d}{d x} [(x - 1) (x - 1)]

$\frac{d}{d x} [(x - 1) (x - 1)]$

Step 2

Expand

(x - 1) (x - 1)

$(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

\frac{d}{d x} [x (x - 1) - 1 (x - 1)]

$\frac{d}{d x} [x (x - 1) - 1 (x - 1)]$

Step 2.2

Apply the distributive property.

\frac{d}{d x} [x \cdot x + x \cdot - 1 - 1 (x - 1)]

$\frac{d}{d x} [x \cdot x + x \cdot - 1 - 1 (x - 1)]$

Step 2.3

Apply the distributive property.

\frac{d}{d x} [x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1]

$\frac{d}{d x} [x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1]$

\frac{d}{d x} [x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1]

$\frac{d}{d x} [x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply

x

$x$ by

x

$x$ .

\frac{d}{d x} [x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1]

$\frac{d}{d x} [x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1]$

Step 3.1.2

Move

- 1

$- 1$ to the left of

x

$x$ .

\frac{d}{d x} [x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1]

$\frac{d}{d x} [x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1]$

Step 3.1.3

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

\frac{d}{d x} [x^{2} - x - 1 x - 1 \cdot - 1]

$\frac{d}{d x} [x^{2} - x - 1 x - 1 \cdot - 1]$

Step 3.1.4

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

\frac{d}{d x} [x^{2} - x - x - 1 \cdot - 1]

$\frac{d}{d x} [x^{2} - x - x - 1 \cdot - 1]$

Step 3.1.5

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{d}{d x} [x^{2} - x - x + 1]

$\frac{d}{d x} [x^{2} - x - x + 1]$

\frac{d}{d x} [x^{2} - x - x + 1]

$\frac{d}{d x} [x^{2} - x - x + 1]$

Step 3.2

Subtract

x

$x$ from

- x

$- x$ .

\frac{d}{d x} [x^{2} - 2 x + 1]

$\frac{d}{d x} [x^{2} - 2 x + 1]$

\frac{d}{d x} [x^{2} - 2 x + 1]

$\frac{d}{d x} [x^{2} - 2 x + 1]$

Step 4

By the Sum Rule, the derivative of

x^{2} - 2 x + 1

$x^{2} - 2 x + 1$ with respect to

x

$x$ is

\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]$ .

\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]$

Step 5

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]

$2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]$

Step 6

Since

- 2

$- 2$ is constant with respect to

x

$x$ , the derivative of

- 2 x

$- 2 x$ with respect to

x

$x$ is

- 2 \frac{d}{d x} [x]

$- 2 \frac{d}{d x} [x]$ .

2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [1]

$2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 7

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

2 x - 2 \cdot 1 + \frac{d}{d x} [1]

$2 x - 2 \cdot 1 + \frac{d}{d x} [1]$

Step 8

Multiply

- 2

$- 2$ by

1

$1$ .

2 x - 2 + \frac{d}{d x} [1]

$2 x - 2 + \frac{d}{d x} [1]$

Step 9

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

2 x - 2 + 0

$2 x - 2 + 0$

Step 10

Add

2 x - 2

$2 x - 2$ and

0

$0$ .

2 x - 2

$2 x - 2$