Calculus Examples

${(x - 7)}^{2}$

Step 1

Rewrite ${(x - 7)}^{2}$ as $(x - 7) (x - 7)$ .

$\frac{d}{d x} [(x - 7) (x - 7)]$

Step 2

Expand $(x - 7) (x - 7)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [x (x - 7) - 7 (x - 7)]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [x \cdot x + x \cdot - 7 - 7 (x - 7)]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [x \cdot x + x \cdot - 7 - 7 x - 7 \cdot - 7]$

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 $x$ by $x$ .

$\frac{d}{d x} [x^{2} + x \cdot - 7 - 7 x - 7 \cdot - 7]$

Step 3.1.2

Move $- 7$ to the left of $x$ .

$\frac{d}{d x} [x^{2} - 7 \cdot x - 7 x - 7 \cdot - 7]$

Step 3.1.3

Multiply $- 7$ by $- 7$ .

$\frac{d}{d x} [x^{2} - 7 x - 7 x + 49]$

Step 3.2

Subtract $7 x$ from $- 7 x$ .

$\frac{d}{d x} [x^{2} - 14 x + 49]$

Step 4

By the Sum Rule, the derivative of $x^{2} - 14 x + 49$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 14 x] + \frac{d}{d x} [49]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 14 x] + \frac{d}{d x} [49]$

Step 5

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

$2 x + \frac{d}{d x} [- 14 x] + \frac{d}{d x} [49]$

Step 6

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x$ with respect to $x$ is $- 14 \frac{d}{d x} [x]$ .

$2 x - 14 \frac{d}{d x} [x] + \frac{d}{d x} [49]$

Step 7

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

$2 x - 14 \cdot 1 + \frac{d}{d x} [49]$

Step 8

Multiply $- 14$ by $1$ .

$2 x - 14 + \frac{d}{d x} [49]$

Step 9

Since $49$ is constant with respect to $x$ , the derivative of $49$ with respect to $x$ is $0$ .

$2 x - 14 + 0$

Step 10

Add $2 x - 14$ and $0$ .

$2 x - 14$