Calculus Examples

$y = x {(6 - 2 x)}^{2}$

Step 1

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

$\frac{d}{d x} [x ((6 - 2 x) (6 - 2 x))]$

Step 2

Expand $(6 - 2 x) (6 - 2 x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [x (6 (6 - 2 x) - 2 x (6 - 2 x))]$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $6$ by $6$ .

$\frac{d}{d x} [x (36 + 6 (- 2 x) - 2 x \cdot 6 - 2 x (- 2 x))]$

Step 3.1.2

Multiply $- 2$ by $6$ .

$\frac{d}{d x} [x (36 - 12 x - 2 x \cdot 6 - 2 x (- 2 x))]$

Step 3.1.3

Multiply $6$ by $- 2$ .

$\frac{d}{d x} [x (36 - 12 x - 12 x - 2 x (- 2 x))]$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [x (36 - 12 x - 12 x - 2 \cdot - 2 x \cdot x)]$

Step 3.1.5

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

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Step 3.1.5.1

Move $x$ .

$\frac{d}{d x} [x (36 - 12 x - 12 x - 2 \cdot - 2 (x \cdot x))]$

Step 3.1.5.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [x (36 - 12 x - 12 x - 2 \cdot - 2 x^{2})]$

Step 3.1.6

Multiply $- 2$ by $- 2$ .

$\frac{d}{d x} [x (36 - 12 x - 12 x + 4 x^{2})]$

Step 3.2

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

$\frac{d}{d x} [x (36 - 24 x + 4 x^{2})]$

Step 4

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 $f (x) = x$ and $g (x) = 36 - 24 x + 4 x^{2}$ .

$x \frac{d}{d x} [36 - 24 x + 4 x^{2}] + (36 - 24 x + 4 x^{2}) \frac{d}{d x} [x]$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $36 - 24 x + 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [36] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [4 x^{2}]$ .

$x (\frac{d}{d x} [36] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [4 x^{2}]) + (36 - 24 x + 4 x^{2}) \frac{d}{d x} [x]$

Step 5.2

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

$x (0 + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [4 x^{2}]) + (36 - 24 x + 4 x^{2}) \frac{d}{d x} [x]$

Step 5.3

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

$x (\frac{d}{d x} [- 24 x] + \frac{d}{d x} [4 x^{2}]) + (36 - 24 x + 4 x^{2}) \frac{d}{d x} [x]$

Step 5.4

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

$x (- 24 \frac{d}{d x} [x] + \frac{d}{d x} [4 x^{2}]) + (36 - 24 x + 4 x^{2}) \frac{d}{d x} [x]$

Step 5.5

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

$x (- 24 \cdot 1 + \frac{d}{d x} [4 x^{2}]) + (36 - 24 x + 4 x^{2}) \frac{d}{d x} [x]$

Step 5.6

Multiply $- 24$ by $1$ .

$x (- 24 + \frac{d}{d x} [4 x^{2}]) + (36 - 24 x + 4 x^{2}) \frac{d}{d x} [x]$

Step 5.7

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$x (- 24 + 4 \frac{d}{d x} [x^{2}]) + (36 - 24 x + 4 x^{2}) \frac{d}{d x} [x]$

Step 5.8

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

$x (- 24 + 4 (2 x)) + (36 - 24 x + 4 x^{2}) \frac{d}{d x} [x]$

Step 5.9

Multiply $2$ by $4$ .

$x (- 24 + 8 x) + (36 - 24 x + 4 x^{2}) \frac{d}{d x} [x]$

Step 5.10

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

$x (- 24 + 8 x) + (36 - 24 x + 4 x^{2}) \cdot 1$

Step 5.11

Multiply $36 - 24 x + 4 x^{2}$ by $1$ .

$x (- 24 + 8 x) + 36 - 24 x + 4 x^{2}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$x \cdot - 24 + x (8 x) + 36 - 24 x + 4 x^{2}$

Step 6.2

Combine terms.

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Step 6.2.1

Move $- 24$ to the left of $x$ .

$- 24 \cdot x + x (8 x) + 36 - 24 x + 4 x^{2}$

Step 6.2.2

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

$- 24 x + 8 (x^{1} x) + 36 - 24 x + 4 x^{2}$

Step 6.2.3

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

$- 24 x + 8 (x^{1} x^{1}) + 36 - 24 x + 4 x^{2}$

Step 6.2.4

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

$- 24 x + 8 x^{1 + 1} + 36 - 24 x + 4 x^{2}$

Step 6.2.5

Add $1$ and $1$ .

$- 24 x + 8 x^{2} + 36 - 24 x + 4 x^{2}$

Step 6.2.6

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

$- 48 x + 8 x^{2} + 36 + 4 x^{2}$

Step 6.2.7

Add $8 x^{2}$ and $4 x^{2}$ .

$- 48 x + 12 x^{2} + 36$

Step 6.3

Reorder terms.

$12 x^{2} - 48 x + 36$