Calculus Examples

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

Step 1

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

$\frac{d}{d x} [2 ((x - 3) (x - 3)) (4 x - 3)]$

Step 2

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

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [2 (x (x - 3) - 3 (x - 3)) (4 x - 3)]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [2 (x \cdot x + x \cdot - 3 - 3 (x - 3)) (4 x - 3)]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [2 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) (4 x - 3)]$

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

Multiply $x$ by $x$ .

$\frac{d}{d x} [2 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) (4 x - 3)]$

Step 3.1.2

Move $- 3$ to the left of $x$ .

$\frac{d}{d x} [2 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) (4 x - 3)]$

Step 3.1.3

Multiply $- 3$ by $- 3$ .

$\frac{d}{d x} [2 (x^{2} - 3 x - 3 x + 9) (4 x - 3)]$

Step 3.2

Subtract $3 x$ from $- 3 x$ .

$\frac{d}{d x} [2 (x^{2} - 6 x + 9) (4 x - 3)]$

Step 4

Since $2$ is constant with respect to $x$ , the derivative of $2 (x^{2} - 6 x + 9) (4 x - 3)$ with respect to $x$ is $2 \frac{d}{d x} [(x^{2} - 6 x + 9) (4 x - 3)]$ .

$2 \frac{d}{d x} [(x^{2} - 6 x + 9) (4 x - 3)]$

Step 5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} - 6 x + 9$ and $g (x) = 4 x - 3$ .

$2 ((x^{2} - 6 x + 9) \frac{d}{d x} [4 x - 3] + (4 x - 3) \frac{d}{d x} [x^{2} - 6 x + 9])$

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $4 x - 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$ .

$2 ((x^{2} - 6 x + 9) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]) + (4 x - 3) \frac{d}{d x} [x^{2} - 6 x + 9])$

Step 6.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$2 ((x^{2} - 6 x + 9) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) + (4 x - 3) \frac{d}{d x} [x^{2} - 6 x + 9])$

Step 6.3

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

$2 ((x^{2} - 6 x + 9) (4 \cdot 1 + \frac{d}{d x} [- 3]) + (4 x - 3) \frac{d}{d x} [x^{2} - 6 x + 9])$

Step 6.4

Multiply $4$ by $1$ .

$2 ((x^{2} - 6 x + 9) (4 + \frac{d}{d x} [- 3]) + (4 x - 3) \frac{d}{d x} [x^{2} - 6 x + 9])$

Step 6.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$2 ((x^{2} - 6 x + 9) (4 + 0) + (4 x - 3) \frac{d}{d x} [x^{2} - 6 x + 9])$

Step 6.6

Simplify the expression.

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Step 6.6.1

Add $4$ and $0$ .

$2 ((x^{2} - 6 x + 9) \cdot 4 + (4 x - 3) \frac{d}{d x} [x^{2} - 6 x + 9])$

Step 6.6.2

Move $4$ to the left of $x^{2} - 6 x + 9$ .

$2 (4 \cdot (x^{2} - 6 x + 9) + (4 x - 3) \frac{d}{d x} [x^{2} - 6 x + 9])$

Step 6.7

By the Sum Rule, the derivative of $x^{2} - 6 x + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]$ .

$2 (4 (x^{2} - 6 x + 9) + (4 x - 3) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]))$

Step 6.8

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

$2 (4 (x^{2} - 6 x + 9) + (4 x - 3) (2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]))$

Step 6.9

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$2 (4 (x^{2} - 6 x + 9) + (4 x - 3) (2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [9]))$

Step 6.10

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

$2 (4 (x^{2} - 6 x + 9) + (4 x - 3) (2 x - 6 \cdot 1 + \frac{d}{d x} [9]))$

Step 6.11

Multiply $- 6$ by $1$ .

$2 (4 (x^{2} - 6 x + 9) + (4 x - 3) (2 x - 6 + \frac{d}{d x} [9]))$

Step 6.12

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

$2 (4 (x^{2} - 6 x + 9) + (4 x - 3) (2 x - 6 + 0))$

Step 6.13

Add $2 x - 6$ and $0$ .

$2 (4 (x^{2} - 6 x + 9) + (4 x - 3) (2 x - 6))$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$2 (4 x^{2} + 4 (- 6 x) + 4 \cdot 9 + (4 x - 3) (2 x - 6))$

Step 7.2

Apply the distributive property.

$2 (4 x^{2}) + 2 (4 (- 6 x)) + 2 (4 \cdot 9) + 2 ((4 x - 3) (2 x - 6))$

Step 7.3

Apply the distributive property.

$2 (4 x^{2}) + 2 (4 (- 6 x)) + 2 (4 \cdot 9) + 2 ((4 x - 3) (2 x) + (4 x - 3) \cdot - 6)$

Step 7.4

Apply the distributive property.

$2 (4 x^{2}) + 2 (4 (- 6 x)) + 2 (4 \cdot 9) + 2 (4 x (2 x) - 3 (2 x) + (4 x - 3) \cdot - 6)$

Step 7.5

Apply the distributive property.

$2 (4 x^{2}) + 2 (4 (- 6 x)) + 2 (4 \cdot 9) + 2 (4 x (2 x) - 3 (2 x) + 4 x \cdot - 6 - 3 \cdot - 6)$

Step 7.6

Apply the distributive property.

$2 (4 x^{2}) + 2 (4 (- 6 x)) + 2 (4 \cdot 9) + 2 (4 x (2 x)) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7

Combine terms.

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Step 7.7.1

Multiply $4$ by $2$ .

$8 x^{2} + 2 (4 (- 6 x)) + 2 (4 \cdot 9) + 2 (4 x (2 x)) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.2

Multiply $- 6$ by $4$ .

$8 x^{2} + 2 (- 24 x) + 2 (4 \cdot 9) + 2 (4 x (2 x)) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.3

Multiply $- 24$ by $2$ .

$8 x^{2} - 48 x + 2 (4 \cdot 9) + 2 (4 x (2 x)) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.4

Multiply $4$ by $9$ .

$8 x^{2} - 48 x + 2 \cdot 36 + 2 (4 x (2 x)) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.5

Multiply $2$ by $36$ .

$8 x^{2} - 48 x + 72 + 2 (4 x (2 x)) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.6

Multiply $2$ by $4$ .

$8 x^{2} - 48 x + 72 + 2 (8 x \cdot x) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.7

Raise $x$ to the power of $1$ .

$8 x^{2} - 48 x + 72 + 2 (8 (x^{1} x)) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.8

Raise $x$ to the power of $1$ .

$8 x^{2} - 48 x + 72 + 2 (8 (x^{1} x^{1})) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$8 x^{2} - 48 x + 72 + 2 (8 x^{1 + 1}) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.10

Add $1$ and $1$ .

$8 x^{2} - 48 x + 72 + 2 (8 x^{2}) + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.11

Multiply $8$ by $2$ .

$8 x^{2} - 48 x + 72 + 16 x^{2} + 2 (- 3 (2 x)) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.12

Multiply $2$ by $- 3$ .

$8 x^{2} - 48 x + 72 + 16 x^{2} + 2 (- 6 x) + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.13

Multiply $- 6$ by $2$ .

$8 x^{2} - 48 x + 72 + 16 x^{2} - 12 x + 2 (4 x \cdot - 6) + 2 (- 3 \cdot - 6)$

Step 7.7.14

Multiply $- 6$ by $4$ .

$8 x^{2} - 48 x + 72 + 16 x^{2} - 12 x + 2 (- 24 x) + 2 (- 3 \cdot - 6)$

Step 7.7.15

Multiply $- 24$ by $2$ .

$8 x^{2} - 48 x + 72 + 16 x^{2} - 12 x - 48 x + 2 (- 3 \cdot - 6)$

Step 7.7.16

Multiply $- 3$ by $- 6$ .

$8 x^{2} - 48 x + 72 + 16 x^{2} - 12 x - 48 x + 2 \cdot 18$

Step 7.7.17

Multiply $2$ by $18$ .

$8 x^{2} - 48 x + 72 + 16 x^{2} - 12 x - 48 x + 36$

Step 7.7.18

Subtract $48 x$ from $- 12 x$ .

$8 x^{2} - 48 x + 72 + 16 x^{2} - 60 x + 36$

Step 7.7.19

Add $8 x^{2}$ and $16 x^{2}$ .

$24 x^{2} - 48 x + 72 - 60 x + 36$

Step 7.7.20

Subtract $60 x$ from $- 48 x$ .

$24 x^{2} - 108 x + 72 + 36$

Step 7.7.21

Add $72$ and $36$ .

$24 x^{2} - 108 x + 108$