Calculus Examples

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

Step 1

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

$\frac{d}{d x} [(2 - 7 x) (2 - 7 x)]$

Step 2

Expand $(2 - 7 x) (2 - 7 x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [2 (2 - 7 x) - 7 x (2 - 7 x)]$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $2$ by $2$ .

$\frac{d}{d x} [4 + 2 (- 7 x) - 7 x \cdot 2 - 7 x (- 7 x)]$

Step 3.1.2

Multiply $- 7$ by $2$ .

$\frac{d}{d x} [4 - 14 x - 7 x \cdot 2 - 7 x (- 7 x)]$

Step 3.1.3

Multiply $2$ by $- 7$ .

$\frac{d}{d x} [4 - 14 x - 14 x - 7 x (- 7 x)]$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [4 - 14 x - 14 x - 7 \cdot - 7 x \cdot x]$

Step 3.1.5

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

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Step 3.1.5.1

Move $x$ .

$\frac{d}{d x} [4 - 14 x - 14 x - 7 \cdot - 7 (x \cdot x)]$

Step 3.1.5.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [4 - 14 x - 14 x - 7 \cdot - 7 x^{2}]$

Step 3.1.6

Multiply $- 7$ by $- 7$ .

$\frac{d}{d x} [4 - 14 x - 14 x + 49 x^{2}]$

Step 3.2

Subtract $14 x$ from $- 14 x$ .

$\frac{d}{d x} [4 - 28 x + 49 x^{2}]$

Step 4

By the Sum Rule, the derivative of $4 - 28 x + 49 x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 28 x] + \frac{d}{d x} [49 x^{2}]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- 28 x] + \frac{d}{d x} [49 x^{2}]$

Step 5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 28 x] + \frac{d}{d x} [49 x^{2}]$

Step 6

Add $0$ and $\frac{d}{d x} [- 28 x]$ .

$\frac{d}{d x} [- 28 x] + \frac{d}{d x} [49 x^{2}]$

Step 7

Since $- 28$ is constant with respect to $x$ , the derivative of $- 28 x$ with respect to $x$ is $- 28 \frac{d}{d x} [x]$ .

$- 28 \frac{d}{d x} [x] + \frac{d}{d x} [49 x^{2}]$

Step 8

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

$- 28 \cdot 1 + \frac{d}{d x} [49 x^{2}]$

Step 9

Multiply $- 28$ by $1$ .

$- 28 + \frac{d}{d x} [49 x^{2}]$

Step 10

Since $49$ is constant with respect to $x$ , the derivative of $49 x^{2}$ with respect to $x$ is $49 \frac{d}{d x} [x^{2}]$ .

$- 28 + 49 \frac{d}{d x} [x^{2}]$

Step 11

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

$- 28 + 49 (2 x)$

Step 12

Simplify the expression.

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Step 12.1

Multiply $2$ by $49$ .

$- 28 + 98 x$

Step 12.2

Reorder terms.

$98 x - 28$