Calculus Examples

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

Step 1

Find the derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.3.2

Subtract $x$ from $- x$ .

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

Step 1.4

Since $2$ is constant with respect to $x$ , the derivative of $2 (x^{2} - 2 x + 1)$ with respect to $x$ is $2 \frac{d}{d x} [x^{2} - 2 x + 1]$ .

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

Step 1.5

By the Sum Rule, the derivative of $x^{2} - 2 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]$ .

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

Step 1.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 1.7

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

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

Step 1.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 1.9

Multiply $- 2$ by $1$ .

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

Step 1.10

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 (2 x - 2 + 0)$

Step 1.11

Add $2 x - 2$ and $0$ .

$2 (2 x - 2)$

Step 1.12

Simplify.

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Step 1.12.1

Apply the distributive property.

$2 (2 x) + 2 \cdot - 2$

Step 1.12.2

Combine terms.

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Step 1.12.2.1

Multiply $2$ by $2$ .

$4 x + 2 \cdot - 2$

Step 1.12.2.2

Multiply $2$ by $- 2$ .

$4 x - 4$

Step 2

Set the derivative equal to $0$ then solve the equation $4 x - 4 = 0$ .

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Step 2.1

Add $4$ to both sides of the equation.

$4 x = 4$

Step 2.2

Divide each term in $4 x = 4$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 x = 4$ by $4$ .

$\frac{4 x}{4} = \frac{4}{4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{4}{4}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $4$ by $4$ .

$x = 1$

Step 3

Solve the original function $f (x) = 2 {(x - 1)}^{2}$ at $x = 1$ .

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Step 3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 2 {((1) - 1)}^{2}$

Step 3.2

Simplify the result.

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Step 3.2.1

Subtract $1$ from $1$ .

$f (1) = 2 \cdot 0^{2}$

Step 3.2.2

Raising $0$ to any positive power yields $0$ .

$f (1) = 2 \cdot 0$

Step 3.2.3

Multiply $2$ by $0$ .

$f (1) = 0$

Step 3.2.4

The final answer is $0$ .

$0$

Step 4

The horizontal tangent line on function $f (x) = 2 {(x - 1)}^{2}$ is $y = 0$ .

$y = 0$

Step 5