Calculus Examples

$f (x) = {(4 - 3 x)}^{2}$

Step 1

Rewrite ${(4 - 3 x)}^{2}$ as $(4 - 3 x) (4 - 3 x)$ .

$\frac{d}{d x} [(4 - 3 x) (4 - 3 x)]$

Step 2

Expand $(4 - 3 x) (4 - 3 x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [4 (4 - 3 x) - 3 x (4 - 3 x)]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [4 \cdot 4 + 4 (- 3 x) - 3 x (4 - 3 x)]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [4 \cdot 4 + 4 (- 3 x) - 3 x \cdot 4 - 3 x (- 3 x)]$

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 $4$ by $4$ .

$\frac{d}{d x} [16 + 4 (- 3 x) - 3 x \cdot 4 - 3 x (- 3 x)]$

Step 3.1.2

Multiply $- 3$ by $4$ .

$\frac{d}{d x} [16 - 12 x - 3 x \cdot 4 - 3 x (- 3 x)]$

Step 3.1.3

Multiply $4$ by $- 3$ .

$\frac{d}{d x} [16 - 12 x - 12 x - 3 x (- 3 x)]$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [16 - 12 x - 12 x - 3 \cdot - 3 x \cdot x]$

Step 3.1.5

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

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Step 3.1.5.1

Move $x$ .

$\frac{d}{d x} [16 - 12 x - 12 x - 3 \cdot - 3 (x \cdot x)]$

Step 3.1.5.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [16 - 12 x - 12 x - 3 \cdot - 3 x^{2}]$

Step 3.1.6

Multiply $- 3$ by $- 3$ .

$\frac{d}{d x} [16 - 12 x - 12 x + 9 x^{2}]$

Step 3.2

Subtract $12 x$ from $- 12 x$ .

$\frac{d}{d x} [16 - 24 x + 9 x^{2}]$

Step 4

By the Sum Rule, the derivative of $16 - 24 x + 9 x^{2}$ with respect to $x$ is $\frac{d}{d x} [16] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9 x^{2}]$ .

$\frac{d}{d x} [16] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9 x^{2}]$

Step 5

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9 x^{2}]$

Step 6

Add $0$ and $\frac{d}{d x} [- 24 x]$ .

$\frac{d}{d x} [- 24 x] + \frac{d}{d x} [9 x^{2}]$

Step 7

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x$ with respect to $x$ is $- 24 \frac{d}{d x} [x]$ .

$- 24 \frac{d}{d x} [x] + \frac{d}{d x} [9 x^{2}]$

Step 8

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

$- 24 \cdot 1 + \frac{d}{d x} [9 x^{2}]$

Step 9

Multiply $- 24$ by $1$ .

$- 24 + \frac{d}{d x} [9 x^{2}]$

Step 10

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$- 24 + 9 \frac{d}{d x} [x^{2}]$

Step 11

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

$- 24 + 9 (2 x)$

Step 12

Simplify the expression.

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Step 12.1

Multiply $2$ by $9$ .

$- 24 + 18 x$

Step 12.2

Reorder terms.

$18 x - 24$