Calculus Examples

$y = 4 x {(x - 2)}^{2}$

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $x$ .

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

Step 3.1.2

Move $- 2$ to the left of $x$ .

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

Step 3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 3.2

Subtract $2 x$ from $- 2 x$ .

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

Step 4

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

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

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

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

Step 6

Differentiate.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Multiply $- 4$ by $1$ .

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

Step 6.6

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 6.7

Add $2 x - 4$ and $0$ .

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

Step 6.8

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

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

Step 6.9

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

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Combine terms.

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

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

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

Step 7.3.4

Add $1$ and $1$ .

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

Step 7.3.5

Multiply $2$ by $4$ .

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

Step 7.3.6

Move $- 4$ to the left of $x$ .

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

Step 7.3.7

Multiply $- 4$ by $4$ .

$8 x^{2} - 16 x + 4 x^{2} + 4 (- 4 x) + 4 \cdot 4$

Step 7.3.8

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

$12 x^{2} - 16 x + 4 (- 4 x) + 4 \cdot 4$

Step 7.3.9

Multiply $- 4$ by $4$ .

$12 x^{2} - 16 x - 16 x + 4 \cdot 4$

Step 7.3.10

Subtract $16 x$ from $- 16 x$ .

$12 x^{2} - 32 x + 4 \cdot 4$

Step 7.3.11

Multiply $4$ by $4$ .

$12 x^{2} - 32 x + 16$