Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 1.3.1.2

Multiply $- x$ by $1$ .

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

Step 1.3.1.3

Multiply $- 1$ by $1$ .

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

Step 1.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.5.1

Move $x$ .

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

Step 1.3.1.5.2

Multiply $x$ by $x$ .

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

Step 1.3.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.7

Multiply $x^{2}$ by $1$ .

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

Step 1.3.2

Subtract $x$ from $- x$ .

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

Step 1.4

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

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

Step 1.5

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

$0 + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [x^{2}]$

Step 1.6

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Multiply $- 2$ by $1$ .

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

Step 1.10

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

$- 2 + 2 x$

Step 1.11

Reorder terms.

$f' (x) = 2 x - 2$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $1$ .

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

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 + 0$

Step 2.3.2

Add $2$ and $0$ .

$f'' (x) = 2$

Step 3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f''' (x) = 0$

Step 4

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

$f^{4} (x) = 0$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $0$ .

$0$