Algebra Examples

$15 {(5 x - 3)}^{2}$

Step 1

Rewrite ${(5 x - 3)}^{2}$ as $(5 x - 3) (5 x - 3)$ .

$\frac{d}{d x} [15 ((5 x - 3) (5 x - 3))]$

Step 2

Expand $(5 x - 3) (5 x - 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [15 (5 x (5 x - 3) - 3 (5 x - 3))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [15 (5 x (5 x) + 5 x \cdot - 3 - 3 (5 x - 3))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [15 (5 x (5 x) + 5 x \cdot - 3 - 3 (5 x) - 3 \cdot - 3)]$

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

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [15 (5 \cdot 5 x \cdot x + 5 x \cdot - 3 - 3 (5 x) - 3 \cdot - 3)]$

Step 3.1.2

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

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Step 3.1.2.1

Move $x$ .

$\frac{d}{d x} [15 (5 \cdot 5 (x \cdot x) + 5 x \cdot - 3 - 3 (5 x) - 3 \cdot - 3)]$

Step 3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [15 (5 \cdot 5 x^{2} + 5 x \cdot - 3 - 3 (5 x) - 3 \cdot - 3)]$

Step 3.1.3

Multiply $5$ by $5$ .

$\frac{d}{d x} [15 (25 x^{2} + 5 x \cdot - 3 - 3 (5 x) - 3 \cdot - 3)]$

Step 3.1.4

Multiply $- 3$ by $5$ .

$\frac{d}{d x} [15 (25 x^{2} - 15 x - 3 (5 x) - 3 \cdot - 3)]$

Step 3.1.5

Multiply $5$ by $- 3$ .

$\frac{d}{d x} [15 (25 x^{2} - 15 x - 15 x - 3 \cdot - 3)]$

Step 3.1.6

Multiply $- 3$ by $- 3$ .

$\frac{d}{d x} [15 (25 x^{2} - 15 x - 15 x + 9)]$

Step 3.2

Subtract $15 x$ from $- 15 x$ .

$\frac{d}{d x} [15 (25 x^{2} - 30 x + 9)]$

Step 4

Since $15$ is constant with respect to $x$ , the derivative of $15 (25 x^{2} - 30 x + 9)$ with respect to $x$ is $15 \frac{d}{d x} [25 x^{2} - 30 x + 9]$ .

$15 \frac{d}{d x} [25 x^{2} - 30 x + 9]$

Step 5

By the Sum Rule, the derivative of $25 x^{2} - 30 x + 9$ with respect to $x$ is $\frac{d}{d x} [25 x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [9]$ .

$15 (\frac{d}{d x} [25 x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [9])$

Step 6

Since $25$ is constant with respect to $x$ , the derivative of $25 x^{2}$ with respect to $x$ is $25 \frac{d}{d x} [x^{2}]$ .

$15 (25 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [9])$

Step 7

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

$15 (25 (2 x) + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [9])$

Step 8

Multiply $2$ by $25$ .

$15 (50 x + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [9])$

Step 9

Since $- 30$ is constant with respect to $x$ , the derivative of $- 30 x$ with respect to $x$ is $- 30 \frac{d}{d x} [x]$ .

$15 (50 x - 30 \frac{d}{d x} [x] + \frac{d}{d x} [9])$

Step 10

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

$15 (50 x - 30 \cdot 1 + \frac{d}{d x} [9])$

Step 11

Multiply $- 30$ by $1$ .

$15 (50 x - 30 + \frac{d}{d x} [9])$

Step 12

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$15 (50 x - 30 + 0)$

Step 13

Add $50 x - 30$ and $0$ .

$15 (50 x - 30)$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$15 (50 x) + 15 \cdot - 30$

Step 14.2

Combine terms.

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Step 14.2.1

Multiply $50$ by $15$ .

$750 x + 15 \cdot - 30$

Step 14.2.2

Multiply $15$ by $- 30$ .

$750 x - 450$