Calculus Examples

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

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 0$ to find the slope of the tangent line.

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Step 1.1

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 $f (x) = x^{2} - 1$ and $g (x) = x^{2} + x + 1$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $2 x$ and $0$ .

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

Step 1.2.4.2

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.3.4

Simplify the numerator.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

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

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Step 1.3.4.1.1.1

Move $x$ .

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

Step 1.3.4.1.1.2

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

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Step 1.3.4.1.1.2.1

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

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

Step 1.3.4.1.1.2.2

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

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

Step 1.3.4.1.1.3

Add $1$ and $2$ .

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

Step 1.3.4.1.2

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

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Step 1.3.4.1.2.1

Move $x$ .

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

Step 1.3.4.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.4.1.3

Multiply $2$ by $1$ .

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

Step 1.3.4.1.4

Multiply $- 1$ by $- 1$ .

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

Step 1.3.4.1.5

Expand $(- x^{2} + 1) (2 x + 1)$ using the FOIL Method.

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Step 1.3.4.1.5.1

Apply the distributive property.

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

Step 1.3.4.1.5.2

Apply the distributive property.

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

Step 1.3.4.1.5.3

Apply the distributive property.

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

Step 1.3.4.1.6

Simplify each term.

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Step 1.3.4.1.6.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.4.1.6.2

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

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Step 1.3.4.1.6.2.1

Move $x$ .

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

Step 1.3.4.1.6.2.2

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

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Step 1.3.4.1.6.2.2.1

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

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

Step 1.3.4.1.6.2.2.2

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

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

Step 1.3.4.1.6.2.3

Add $1$ and $2$ .

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

Step 1.3.4.1.6.3

Multiply $- 1$ by $2$ .

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

Step 1.3.4.1.6.4

Multiply $- 1$ by $1$ .

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

Step 1.3.4.1.6.5

Multiply $2 x$ by $1$ .

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

Step 1.3.4.1.6.6

Multiply $1$ by $1$ .

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

Step 1.3.4.2

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

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Step 1.3.4.2.1

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

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

Step 1.3.4.2.2

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

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

Step 1.3.4.3

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

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

Step 1.3.4.4

Add $2 x$ and $2 x$ .

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

Step 1.4

Evaluate the derivative at $x = 1$ .

$\frac{{(1)}^{2} + 4 (1) + 1}{{({(1)}^{2} + 1 + 1)}^{2}}$

Step 1.5

Simplify.

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Step 1.5.1

Remove parentheses.

$\frac{{(1)}^{2} + 4 (1) + 1}{{({(1)}^{2} + 1 + 1)}^{2}}$

Step 1.5.2

Simplify the numerator.

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Step 1.5.2.1

One to any power is one.

$\frac{1 + 4 (1) + 1}{{(1^{2} + 1 + 1)}^{2}}$

Step 1.5.2.2

Multiply $4$ by $1$ .

$\frac{1 + 4 + 1}{{(1^{2} + 1 + 1)}^{2}}$

Step 1.5.2.3

Add $1$ and $4$ .

$\frac{5 + 1}{{(1^{2} + 1 + 1)}^{2}}$

Step 1.5.2.4

Add $5$ and $1$ .

$\frac{6}{{(1^{2} + 1 + 1)}^{2}}$

Step 1.5.3

Simplify the denominator.

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Step 1.5.3.1

One to any power is one.

$\frac{6}{{(1 + 1 + 1)}^{2}}$

Step 1.5.3.2

Add $1$ and $1$ .

$\frac{6}{{(2 + 1)}^{2}}$

Step 1.5.3.3

Add $2$ and $1$ .

$\frac{6}{3^{2}}$

Step 1.5.3.4

Raise $3$ to the power of $2$ .

$\frac{6}{9}$

Step 1.5.4

Cancel the common factor of $6$ and $9$ .

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Step 1.5.4.1

Factor $3$ out of $6$ .

$\frac{3 (2)}{9}$

Step 1.5.4.2

Cancel the common factors.

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Step 1.5.4.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 2}{3 \cdot 3}$

Step 1.5.4.2.2

Cancel the common factor.

$\frac{3 \cdot 2}{3 \cdot 3}$

Step 1.5.4.2.3

Rewrite the expression.

$\frac{2}{3}$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $\frac{2}{3}$ and a given point $(1, 0)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = \frac{2}{3} \cdot (x - (1))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y + 0 = \frac{2}{3} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

$y = \frac{2}{3} \cdot (x - 1)$

Step 2.3.2

Simplify $\frac{2}{3} \cdot (x - 1)$ .

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Step 2.3.2.1

Apply the distributive property.

$y = \frac{2}{3} x + \frac{2}{3} \cdot - 1$

Step 2.3.2.2

Combine $\frac{2}{3}$ and $x$ .

$y = \frac{2 x}{3} + \frac{2}{3} \cdot - 1$

Step 2.3.2.3

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 2.3.2.3.1

Combine $\frac{2}{3}$ and $- 1$ .

$y = \frac{2 x}{3} + \frac{2 \cdot - 1}{3}$

Step 2.3.2.3.2

Multiply $2$ by $- 1$ .

$y = \frac{2 x}{3} + \frac{- 2}{3}$

Step 2.3.2.4

Move the negative in front of the fraction.

$y = \frac{2 x}{3} - \frac{2}{3}$

Step 2.3.3

Reorder terms.

$y = \frac{2}{3} x - \frac{2}{3}$

Step 3