Calculus Examples

$f (x) = {(4 x - 6)}^{2}$

Step 1

Rewrite ${(4 x - 6)}^{2}$ as $(4 x - 6) (4 x - 6)$ .

$\frac{d}{d x} [(4 x - 6) (4 x - 6)]$

Step 2

Expand $(4 x - 6) (4 x - 6)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [4 x (4 x - 6) - 6 (4 x - 6)]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [4 x (4 x) + 4 x \cdot - 6 - 6 (4 x - 6)]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [4 x (4 x) + 4 x \cdot - 6 - 6 (4 x) - 6 \cdot - 6]$

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.

$\frac{d}{d x} [4 \cdot 4 x \cdot x + 4 x \cdot - 6 - 6 (4 x) - 6 \cdot - 6]$

Step 3.1.2

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

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Step 3.1.2.1

Move $x$ .

$\frac{d}{d x} [4 \cdot 4 (x \cdot x) + 4 x \cdot - 6 - 6 (4 x) - 6 \cdot - 6]$

Step 3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [4 \cdot 4 x^{2} + 4 x \cdot - 6 - 6 (4 x) - 6 \cdot - 6]$

Step 3.1.3

Multiply $4$ by $4$ .

$\frac{d}{d x} [16 x^{2} + 4 x \cdot - 6 - 6 (4 x) - 6 \cdot - 6]$

Step 3.1.4

Multiply $- 6$ by $4$ .

$\frac{d}{d x} [16 x^{2} - 24 x - 6 (4 x) - 6 \cdot - 6]$

Step 3.1.5

Multiply $4$ by $- 6$ .

$\frac{d}{d x} [16 x^{2} - 24 x - 24 x - 6 \cdot - 6]$

Step 3.1.6

Multiply $- 6$ by $- 6$ .

$\frac{d}{d x} [16 x^{2} - 24 x - 24 x + 36]$

Step 3.2

Subtract $24 x$ from $- 24 x$ .

$\frac{d}{d x} [16 x^{2} - 48 x + 36]$

Step 4

By the Sum Rule, the derivative of $16 x^{2} - 48 x + 36$ with respect to $x$ is $\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [- 48 x] + \frac{d}{d x} [36]$ .

$\frac{d}{d x} [16 x^{2}] + \frac{d}{d x} [- 48 x] + \frac{d}{d x} [36]$

Step 5

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{2}$ with respect to $x$ is $16 \frac{d}{d x} [x^{2}]$ .

$16 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 48 x] + \frac{d}{d x} [36]$

Step 6

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

$16 (2 x) + \frac{d}{d x} [- 48 x] + \frac{d}{d x} [36]$

Step 7

Multiply $2$ by $16$ .

$32 x + \frac{d}{d x} [- 48 x] + \frac{d}{d x} [36]$

Step 8

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48 x$ with respect to $x$ is $- 48 \frac{d}{d x} [x]$ .

$32 x - 48 \frac{d}{d x} [x] + \frac{d}{d x} [36]$

Step 9

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

$32 x - 48 \cdot 1 + \frac{d}{d x} [36]$

Step 10

Multiply $- 48$ by $1$ .

$32 x - 48 + \frac{d}{d x} [36]$

Step 11

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$32 x - 48 + 0$

Step 12

Add $32 x - 48$ and $0$ .

$32 x - 48$