Calculus Examples

$f (x) = {(2 x - 1)}^{2}$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = {(2 (x + h) - 1)}^{2}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify by multiplying through.

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Step 2.1.2.1.1

Apply the distributive property.

$f (x + h) = {(2 x + 2 h - 1)}^{2}$

Step 2.1.2.1.2

Rewrite ${(2 x + 2 h - 1)}^{2}$ as $(2 x + 2 h - 1) (2 x + 2 h - 1)$ .

$f (x + h) = (2 x + 2 h - 1) (2 x + 2 h - 1)$

Step 2.1.2.2

Expand $(2 x + 2 h - 1) (2 x + 2 h - 1)$ by multiplying each term in the first expression by each term in the second expression.

$f (x + h) = 2 x (2 x) + 2 x (2 h) + 2 x \cdot - 1 + 2 h (2 x) + 2 h (2 h) + 2 h \cdot - 1 - 1 (2 x) - 1 (2 h) - 1 \cdot - 1$

Step 2.1.2.3

Simplify each term.

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Step 2.1.2.3.1

Rewrite using the commutative property of multiplication.

$f (x + h) = 2 \cdot (2 x \cdot x) + 2 x (2 h) + 2 x \cdot - 1 + 2 h (2 x) + 2 h (2 h) + 2 h \cdot - 1 - 1 (2 x) - 1 (2 h) - 1 \cdot - 1$

Step 2.1.2.3.2

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

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Step 2.1.2.3.2.1

Move $x$ .

$f (x + h) = 2 \cdot (2 (x \cdot x)) + 2 x (2 h) + 2 x \cdot - 1 + 2 h (2 x) + 2 h (2 h) + 2 h \cdot - 1 - 1 (2 x) - 1 (2 h) - 1 \cdot - 1$

Step 2.1.2.3.2.2

Multiply $x$ by $x$ .

$f (x + h) = 2 \cdot (2 x^{2}) + 2 x (2 h) + 2 x \cdot - 1 + 2 h (2 x) + 2 h (2 h) + 2 h \cdot - 1 - 1 (2 x) - 1 (2 h) - 1 \cdot - 1$

Step 2.1.2.3.3

Multiply $2$ by $2$ .

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

Step 2.1.2.3.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.2.3.5

Multiply $2$ by $2$ .

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

Step 2.1.2.3.6

Multiply $- 1$ by $2$ .

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

Step 2.1.2.3.7

Rewrite using the commutative property of multiplication.

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

Step 2.1.2.3.8

Multiply $2$ by $2$ .

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

Step 2.1.2.3.9

Rewrite using the commutative property of multiplication.

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

Step 2.1.2.3.10

Multiply $h$ by $h$ by adding the exponents.

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Step 2.1.2.3.10.1

Move $h$ .

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

Step 2.1.2.3.10.2

Multiply $h$ by $h$ .

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

Step 2.1.2.3.11

Multiply $2$ by $2$ .

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

Step 2.1.2.3.12

Multiply $- 1$ by $2$ .

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

Step 2.1.2.3.13

Multiply $2$ by $- 1$ .

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

Step 2.1.2.3.14

Multiply $2$ by $- 1$ .

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

Step 2.1.2.3.15

Multiply $- 1$ by $- 1$ .

$f (x + h) = 4 x^{2} + 4 x h - 2 x + 4 h x + 4 h^{2} - 2 h - 2 x - 2 h + 1$

Step 2.1.2.4

Add $4 x h$ and $4 h x$ .

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Step 2.1.2.4.1

Move $x$ .

$f (x + h) = 4 x^{2} + 4 h x + 4 h x - 2 x + 4 h^{2} - 2 h - 2 x - 2 h + 1$

Step 2.1.2.4.2

Add $4 h x$ and $4 h x$ .

$f (x + h) = 4 x^{2} + 8 h x - 2 x + 4 h^{2} - 2 h - 2 x - 2 h + 1$

Step 2.1.2.5

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

$f (x + h) = 4 x^{2} + 8 h x + 4 h^{2} - 2 h - 4 x - 2 h + 1$

Step 2.1.2.6

Subtract $2 h$ from $- 2 h$ .

$f (x + h) = 4 x^{2} + 8 h x + 4 h^{2} - 4 h - 4 x + 1$

Step 2.1.2.7

The final answer is $4 x^{2} + 8 h x + 4 h^{2} - 4 h - 4 x + 1$ .

$4 x^{2} + 8 h x + 4 h^{2} - 4 h - 4 x + 1$

Step 2.2

Reorder.

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Step 2.2.1

Move $4 x^{2}$ .

$8 h x + 4 h^{2} + 4 x^{2} - 4 h - 4 x + 1$

Step 2.2.2

Reorder $8 h x$ and $4 h^{2}$ .

$4 h^{2} + 8 h x + 4 x^{2} - 4 h - 4 x + 1$

Step 2.3

Find the components of the definition.

$f (x + h) = 4 h^{2} + 8 h x + 4 x^{2} - 4 h - 4 x + 1$

$f (x) = 4 x^{2} - 4 x + 1$

$f (x + h) = 4 h^{2} + 8 h x + 4 x^{2} - 4 h - 4 x + 1$

$f (x) = 4 x^{2} - 4 x + 1$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{4 h^{2} + 8 h x + 4 x^{2} - 4 h - 4 x + 1 - (4 x^{2} - 4 x + 1)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{4 h^{2} + 8 h x + 4 x^{2} - 4 h - 4 x + 1 - (4 x^{2}) - (- 4 x) - 1 \cdot 1}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $4$ by $- 1$ .

$\frac{4 h^{2} + 8 h x + 4 x^{2} - 4 h - 4 x + 1 - 4 x^{2} - (- 4 x) - 1 \cdot 1}{h}$

Step 4.1.2.2

Multiply $- 4$ by $- 1$ .

$\frac{4 h^{2} + 8 h x + 4 x^{2} - 4 h - 4 x + 1 - 4 x^{2} + 4 x - 1 \cdot 1}{h}$

Step 4.1.2.3

Multiply $- 1$ by $1$ .

$\frac{4 h^{2} + 8 h x + 4 x^{2} - 4 h - 4 x + 1 - 4 x^{2} + 4 x - 1}{h}$

Step 4.1.3

Subtract $4 x^{2}$ from $4 x^{2}$ .

$\frac{4 h^{2} + 8 h x - 4 h - 4 x + 1 + 0 + 4 x - 1}{h}$

Step 4.1.4

Add $4 h^{2}$ and $0$ .

$\frac{4 h^{2} + 8 h x - 4 h - 4 x + 1 + 4 x - 1}{h}$

Step 4.1.5

Add $- 4 x$ and $4 x$ .

$\frac{4 h^{2} + 8 h x - 4 h + 0 + 1 - 1}{h}$

Step 4.1.6

Add $4 h^{2}$ and $0$ .

$\frac{4 h^{2} + 8 h x - 4 h + 1 - 1}{h}$

Step 4.1.7

Subtract $1$ from $1$ .

$\frac{4 h^{2} + 8 h x - 4 h + 0}{h}$

Step 4.1.8

Add $4 h^{2} + 8 h x - 4 h$ and $0$ .

$\frac{4 h^{2} + 8 h x - 4 h}{h}$

Step 4.1.9

Factor $4 h$ out of $4 h^{2} + 8 h x - 4 h$ .

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Step 4.1.9.1

Factor $4 h$ out of $4 h^{2}$ .

$\frac{4 h \cdot h + 8 h x - 4 h}{h}$

Step 4.1.9.2

Factor $4 h$ out of $8 h x$ .

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

Step 4.1.9.3

Factor $4 h$ out of $- 4 h$ .

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

Step 4.1.9.4

Factor $4 h$ out of $4 h \cdot h + 4 h (2 x)$ .

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

Step 4.1.9.5

Factor $4 h$ out of $4 h (h + 2 x) + 4 h \cdot - 1$ .

$\frac{4 h (h + 2 x - 1)}{h}$

Step 4.2

Simplify terms.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 h (h + 2 x - 1)}{h}$

Step 4.2.1.2

Divide $4 (h + 2 x - 1)$ by $1$ .

$4 (h + 2 x - 1)$

Step 4.2.2

Apply the distributive property.

$4 h + 4 (2 x) + 4 \cdot - 1$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $2$ by $4$ .

$4 h + 8 x + 4 \cdot - 1$

Step 4.3.2

Multiply $4$ by $- 1$ .

$4 h + 8 x - 4$

Step 4.4

Reorder $4 h$ and $8 x$ .

$8 x + 4 h - 4$

Step 5