Calculus Examples

$\frac{x^{2} - x + 1}{x^{2} + x + 1}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} - x + 1$ and $g (x) = x^{2} + x + 1$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [1]$ .

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

Step 2.2

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

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

Step 2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 2.4

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

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

Step 2.5

Multiply $- 1$ by $1$ .

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

Step 2.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(x^{2} + x + 1) (2 x - 1 + 0) - (x^{2} - x + 1) \frac{d}{d x} [x^{2} + x + 1]}{{(x^{2} + x + 1)}^{2}}$

Step 2.7

Add $2 x - 1$ and $0$ .

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

Step 2.8

By the Sum Rule, the derivative of $x^{2} + x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(x^{2} + x + 1) (2 x - 1) - (x^{2} - x + 1) (2 x + 1 + 0)}{{(x^{2} + x + 1)}^{2}}$

Step 2.12

Add $2 x + 1$ and $0$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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

Step 3.2.1.2

Simplify each term.

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Step 3.2.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.2.1.2.2.1

Move $x$ .

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

Step 3.2.1.2.2.2

Multiply $x$ by $x^{2}$ .

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Step 3.2.1.2.2.2.1

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

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

Step 3.2.1.2.2.2.2

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

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

Step 3.2.1.2.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + x^{2} \cdot - 1 + x (2 x) + x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.2.3

Move $- 1$ to the left of $x^{2}$ .

$\frac{2 x^{3} - 1 \cdot x^{2} + x (2 x) + x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.2.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\frac{2 x^{3} - x^{2} + x (2 x) + x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.2.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - x^{2} + 2 x \cdot x + x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.2.6

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

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Step 3.2.1.2.6.1

Move $x$ .

$\frac{2 x^{3} - x^{2} + 2 (x \cdot x) + x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.2.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - x^{2} + 2 x^{2} + x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.2.7

Move $- 1$ to the left of $x$ .

$\frac{2 x^{3} - x^{2} + 2 x^{2} - 1 \cdot x + 1 (2 x) + 1 \cdot - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.2.8

Rewrite $- 1 x$ as $- x$ .

$\frac{2 x^{3} - x^{2} + 2 x^{2} - x + 1 (2 x) + 1 \cdot - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.2.9

Multiply $2 x$ by $1$ .

$\frac{2 x^{3} - x^{2} + 2 x^{2} - x + 2 x + 1 \cdot - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.2.10

Multiply $- 1$ by $1$ .

$\frac{2 x^{3} - x^{2} + 2 x^{2} - x + 2 x - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.3

Add $- x^{2}$ and $2 x^{2}$ .

$\frac{2 x^{3} + x^{2} - x + 2 x - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.4

Add $- x$ and $2 x$ .

$\frac{2 x^{3} + x^{2} + x - 1 + (- x^{2} - - x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.5

Simplify each term.

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Step 3.2.1.5.1

Multiply $- - x$ .

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Step 3.2.1.5.1.1

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{3} + x^{2} + x - 1 + (- x^{2} + 1 x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.5.1.2

Multiply $x$ by $1$ .

$\frac{2 x^{3} + x^{2} + x - 1 + (- x^{2} + x - 1 \cdot 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.5.2

Multiply $- 1$ by $1$ .

$\frac{2 x^{3} + x^{2} + x - 1 + (- x^{2} + x - 1) (2 x + 1)}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.6

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

$\frac{2 x^{3} + x^{2} + x - 1 - x^{2} (2 x) - x^{2} \cdot 1 + x (2 x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7

Simplify each term.

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Step 3.2.1.7.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + x^{2} + x - 1 - 1 \cdot 2 x^{2} x - x^{2} \cdot 1 + x (2 x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.2.1.7.2.1

Move $x$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot 1 + x (2 x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.2.2

Multiply $x$ by $x^{2}$ .

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Step 3.2.1.7.2.2.1

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

$\frac{2 x^{3} + x^{2} + x - 1 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot 1 + x (2 x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.2.2.2

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

$\frac{2 x^{3} + x^{2} + x - 1 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot 1 + x (2 x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 1 \cdot 2 x^{3} - x^{2} \cdot 1 + x (2 x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 2 x^{3} - x^{2} \cdot 1 + x (2 x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.4

Multiply $- 1$ by $1$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 2 x^{3} - x^{2} + x (2 x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + x^{2} + x - 1 - 2 x^{3} - x^{2} + 2 x \cdot x + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.6

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

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Step 3.2.1.7.6.1

Move $x$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 2 x^{3} - x^{2} + 2 (x \cdot x) + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 2 x^{3} - x^{2} + 2 x^{2} + x \cdot 1 - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.7

Multiply $x$ by $1$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 2 x^{3} - x^{2} + 2 x^{2} + x - 1 (2 x) - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.8

Multiply $2$ by $- 1$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 2 x^{3} - x^{2} + 2 x^{2} + x - 2 x - 1 \cdot 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.7.9

Multiply $- 1$ by $1$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 2 x^{3} - x^{2} + 2 x^{2} + x - 2 x - 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.8

Add $- x^{2}$ and $2 x^{2}$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 2 x^{3} + x^{2} + x - 2 x - 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.1.9

Subtract $2 x$ from $x$ .

$\frac{2 x^{3} + x^{2} + x - 1 - 2 x^{3} + x^{2} - x - 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.2

Combine the opposite terms in $2 x^{3} + x^{2} + x - 1 - 2 x^{3} + x^{2} - x - 1$ .

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Step 3.2.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$\frac{x^{2} + x - 1 + 0 + x^{2} - x - 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.2.2

Add $x^{2} + x - 1$ and $0$ .

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

Step 3.2.2.3

Subtract $x$ from $x$ .

$\frac{x^{2} - 1 + x^{2} + 0 - 1}{{(x^{2} + x + 1)}^{2}}$

Step 3.2.2.4

Add $x^{2} - 1 + x^{2}$ and $0$ .

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

Step 3.2.3

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

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

Step 3.2.4

Subtract $1$ from $- 1$ .

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Factor $2$ out of $2 x^{2} - 2$ .

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Step 3.3.1.1

Factor $2$ out of $2 x^{2}$ .

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

Step 3.3.1.2

Factor $2$ out of $- 2$ .

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

Step 3.3.1.3

Factor $2$ out of $2 (x^{2}) + 2 (- 1)$ .

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

Step 3.3.2

Rewrite $1$ as $1^{2}$ .

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

Step 3.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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