Calculus Examples

${(x + 1)}^{3} - x^{3}$

Step 1

By the Sum Rule, the derivative of ${(x + 1)}^{3} - x^{3}$ with respect to $x$ is $\frac{d}{d x} [{(x + 1)}^{3}] + \frac{d}{d x} [- x^{3}]$ .

$\frac{d}{d x} [{(x + 1)}^{3}] + \frac{d}{d x} [- x^{3}]$

Step 2

Evaluate $\frac{d}{d x} [{(x + 1)}^{3}]$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x + 1$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x + 1$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [x + 1] + \frac{d}{d x} [- x^{3}]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d x} [x + 1] + \frac{d}{d x} [- x^{3}]$

Step 2.1.3

Replace all occurrences of $u$ with $x + 1$ .

$3 {(x + 1)}^{2} \frac{d}{d x} [x + 1] + \frac{d}{d x} [- x^{3}]$

Step 2.2

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

$3 {(x + 1)}^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [1]) + \frac{d}{d x} [- x^{3}]$

Step 2.3

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

$3 {(x + 1)}^{2} (1 + \frac{d}{d x} [1]) + \frac{d}{d x} [- x^{3}]$

Step 2.4

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

$3 {(x + 1)}^{2} (1 + 0) + \frac{d}{d x} [- x^{3}]$

Step 2.5

Add $1$ and $0$ .

$3 {(x + 1)}^{2} \cdot 1 + \frac{d}{d x} [- x^{3}]$

Step 2.6

Multiply $3$ by $1$ .

$3 {(x + 1)}^{2} + \frac{d}{d x} [- x^{3}]$

Step 3

Evaluate $\frac{d}{d x} [- x^{3}]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$3 {(x + 1)}^{2} - \frac{d}{d x} [x^{3}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 {(x + 1)}^{2} - (3 x^{2})$

Step 3.3

Multiply $3$ by $- 1$ .

$3 {(x + 1)}^{2} - 3 x^{2}$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$- 3 x^{2} + 3 {(x + 1)}^{2}$

Step 4.2

Simplify each term.

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Step 4.2.1

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

$- 3 x^{2} + 3 ((x + 1) (x + 1))$

Step 4.2.2

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

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Step 4.2.2.1

Apply the distributive property.

$- 3 x^{2} + 3 (x (x + 1) + 1 (x + 1))$

Step 4.2.2.2

Apply the distributive property.

$- 3 x^{2} + 3 (x \cdot x + x \cdot 1 + 1 (x + 1))$

Step 4.2.2.3

Apply the distributive property.

$- 3 x^{2} + 3 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1)$

Step 4.2.3

Simplify and combine like terms.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Multiply $x$ by $x$ .

$- 3 x^{2} + 3 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1)$

Step 4.2.3.1.2

Multiply $x$ by $1$ .

$- 3 x^{2} + 3 (x^{2} + x + 1 x + 1 \cdot 1)$

Step 4.2.3.1.3

Multiply $x$ by $1$ .

$- 3 x^{2} + 3 (x^{2} + x + x + 1 \cdot 1)$

Step 4.2.3.1.4

Multiply $1$ by $1$ .

$- 3 x^{2} + 3 (x^{2} + x + x + 1)$

Step 4.2.3.2

Add $x$ and $x$ .

$- 3 x^{2} + 3 (x^{2} + 2 x + 1)$

Step 4.2.4

Apply the distributive property.

$- 3 x^{2} + 3 x^{2} + 3 (2 x) + 3 \cdot 1$

Step 4.2.5

Simplify.

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Step 4.2.5.1

Multiply $2$ by $3$ .

$- 3 x^{2} + 3 x^{2} + 6 x + 3 \cdot 1$

Step 4.2.5.2

Multiply $3$ by $1$ .

$- 3 x^{2} + 3 x^{2} + 6 x + 3$

Step 4.3

Add $- 3 x^{2}$ and $3 x^{2}$ .

$0 + 6 x + 3$

Step 4.4

Add $0$ and $6 x$ .

$6 x + 3$