Calculus Examples

Let $g (x) = \frac{x^{2} - 3}{x + 2}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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 $f (x) = x^{2} - 3$ and $g (x) = x + 2$ .

$\frac{(x + 2) \frac{d}{d x} [x^{2} - 3] - (x^{2} - 3) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]$ .

$\frac{(x + 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]) - (x^{2} - 3) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.1.2.2

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

$\frac{(x + 2) (2 x + \frac{d}{d x} [- 3]) - (x^{2} - 3) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.1.2.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{(x + 2) (2 x + 0) - (x^{2} - 3) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $2 x$ and $0$ .

$\frac{(x + 2) (2 x) - (x^{2} - 3) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.1.2.4.2

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

$\frac{2 \cdot (x + 2) x - (x^{2} - 3) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.1.2.5

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

$\frac{2 (x + 2) x - (x^{2} - 3) (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 1.1.2.6

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

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

Step 1.1.2.7

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{2 (x + 2) x - (x^{2} - 3) (1 + 0)}{{(x + 2)}^{2}}$

Step 1.1.2.8

Simplify the expression.

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Step 1.1.2.8.1

Add $1$ and $0$ .

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

Step 1.1.2.8.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

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

Step 1.1.3.2

Apply the distributive property.

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

Step 1.1.3.3

Apply the distributive property.

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

Step 1.1.3.4

Simplify the numerator.

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Step 1.1.3.4.1

Simplify each term.

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Step 1.1.3.4.1.1

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

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Step 1.1.3.4.1.1.1

Move $x$ .

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

Step 1.1.3.4.1.1.2

Multiply $x$ by $x$ .

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

Step 1.1.3.4.1.2

Multiply $2$ by $2$ .

$\frac{2 x^{2} + 4 x - x^{2} - - 3}{{(x + 2)}^{2}}$

Step 1.1.3.4.1.3

Multiply $- 1$ by $- 3$ .

$\frac{2 x^{2} + 4 x - x^{2} + 3}{{(x + 2)}^{2}}$

Step 1.1.3.4.2

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

$\frac{x^{2} + 4 x + 3}{{(x + 2)}^{2}}$

Step 1.1.3.5

Factor $x^{2} + 4 x + 3$ using the AC method.

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Step 1.1.3.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Step 1.1.3.5.2

Write the factored form using these integers.

$f' (x) = \frac{(x + 1) (x + 3)}{{(x + 2)}^{2}}$

Step 1.2

The first derivative of $g (x)$ with respect to $x$ is $\frac{(x + 1) (x + 3)}{{(x + 2)}^{2}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{(x + 1) (x + 3)}{{(x + 2)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{(x + 1) (x + 3)}{{(x + 2)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$(x + 1) (x + 3) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x + 3 = 0$

Step 2.3.2

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.2.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.3.3

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.3.3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.3.4

The final solution is all the values that make $(x + 1) (x + 3) = 0$ true.

$x = - 1, - 3$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{(x + 1) (x + 3)}{{(x + 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x + 2)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4

Evaluate $\frac{x^{2} - 3}{x + 2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 1$ .

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Step 4.1.1

Substitute $- 1$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Raise $- 1$ to the power of $2$ .

$\frac{1 - 3}{- 1 + 2}$

Step 4.1.2.1.2

Subtract $3$ from $1$ .

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

Step 4.1.2.2

Simplify the expression.

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Step 4.1.2.2.1

Add $- 1$ and $2$ .

$\frac{- 2}{1}$

Step 4.1.2.2.2

Divide $- 2$ by $1$ .

$- 2$

Step 4.2

Evaluate at $x = - 3$ .

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Step 4.2.1

Substitute $- 3$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

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

$\frac{9 - 3}{- 3 + 2}$

Step 4.2.2.1.2

Subtract $3$ from $9$ .

$\frac{6}{- 3 + 2}$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Add $- 3$ and $2$ .

$\frac{6}{- 1}$

Step 4.2.2.2.2

Divide $6$ by $- 1$ .

$- 6$

Step 4.3

Evaluate at $x = - 2$ .

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Step 4.3.1

Substitute $- 2$ for $x$ .

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

Step 4.3.2

Simplify.

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Step 4.3.2.1

Add $- 2$ and $2$ .

$\frac{{(- 2)}^{2} - 3}{0}$

Step 4.3.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.4

List all of the points.

$(- 1, - 2), (- 3, - 6)$

Step 5