Calculus Examples

$y = \frac{x^{3} - 8}{x - 2}$

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 $f (x) = x^{3} - 8$ and $g (x) = x - 2$ .

$\frac{(x - 2) \frac{d}{d x} [x^{3} - 8] - (x^{3} - 8) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{3} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8]$ .

$\frac{(x - 2) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8]) - (x^{3} - 8) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

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 $n = 3$ .

$\frac{(x - 2) (3 x^{2} + \frac{d}{d x} [- 8]) - (x^{3} - 8) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.2.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$\frac{(x - 2) (3 x^{2} + 0) - (x^{3} - 8) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $3 x^{2}$ and $0$ .

$\frac{(x - 2) (3 x^{2}) - (x^{3} - 8) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.2.4.2

Move $3$ to the left of $x - 2$ .

$\frac{3 \cdot (x - 2) x^{2} - (x^{3} - 8) \frac{d}{d x} [x - 2]}{{(x - 2)}^{2}}$

Step 1.1.2.5

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

$\frac{3 (x - 2) x^{2} - (x^{3} - 8) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}$

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 $n = 1$ .

$\frac{3 (x - 2) x^{2} - (x^{3} - 8) (1 + \frac{d}{d x} [- 2])}{{(x - 2)}^{2}}$

Step 1.1.2.7

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

$\frac{3 (x - 2) x^{2} - (x^{3} - 8) (1 + 0)}{{(x - 2)}^{2}}$

Step 1.1.2.8

Simplify the expression.

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Step 1.1.2.8.1

Add $1$ and $0$ .

$\frac{3 (x - 2) x^{2} - (x^{3} - 8) \cdot 1}{{(x - 2)}^{2}}$

Step 1.1.2.8.2

Multiply $- 1$ by $1$ .

$\frac{3 (x - 2) x^{2} - (x^{3} - 8)}{{(x - 2)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{(3 x + 3 \cdot - 2) x^{2} - (x^{3} - 8)}{{(x - 2)}^{2}}$

Step 1.1.3.2

Apply the distributive property.

$\frac{3 x \cdot x^{2} + 3 \cdot - 2 x^{2} - (x^{3} - 8)}{{(x - 2)}^{2}}$

Step 1.1.3.3

Apply the distributive property.

$\frac{3 x \cdot x^{2} + 3 \cdot - 2 x^{2} - x^{3} - - 8}{{(x - 2)}^{2}}$

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

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Step 1.1.3.4.1.1.1

Move $x^{2}$ .

$\frac{3 (x^{2} x) + 3 \cdot - 2 x^{2} - x^{3} - - 8}{{(x - 2)}^{2}}$

Step 1.1.3.4.1.1.2

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

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Step 1.1.3.4.1.1.2.1

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

$\frac{3 (x^{2} x^{1}) + 3 \cdot - 2 x^{2} - x^{3} - - 8}{{(x - 2)}^{2}}$

Step 1.1.3.4.1.1.2.2

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

$\frac{3 x^{2 + 1} + 3 \cdot - 2 x^{2} - x^{3} - - 8}{{(x - 2)}^{2}}$

Step 1.1.3.4.1.1.3

Add $2$ and $1$ .

$\frac{3 x^{3} + 3 \cdot - 2 x^{2} - x^{3} - - 8}{{(x - 2)}^{2}}$

Step 1.1.3.4.1.2

Multiply $3$ by $- 2$ .

$\frac{3 x^{3} - 6 x^{2} - x^{3} - - 8}{{(x - 2)}^{2}}$

Step 1.1.3.4.1.3

Multiply $- 1$ by $- 8$ .

$\frac{3 x^{3} - 6 x^{2} - x^{3} + 8}{{(x - 2)}^{2}}$

Step 1.1.3.4.2

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

$\frac{2 x^{3} - 6 x^{2} + 8}{{(x - 2)}^{2}}$

Step 1.1.3.5

Factor $2$ out of $2 x^{3} - 6 x^{2} + 8$ .

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Step 1.1.3.5.1

Factor $2$ out of $2 x^{3}$ .

$\frac{2 (x^{3}) - 6 x^{2} + 8}{{(x - 2)}^{2}}$

Step 1.1.3.5.2

Factor $2$ out of $- 6 x^{2}$ .

$\frac{2 (x^{3}) + 2 (- 3 x^{2}) + 8}{{(x - 2)}^{2}}$

Step 1.1.3.5.3

Factor $2$ out of $8$ .

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

Step 1.1.3.5.4

Factor $2$ out of $2 x^{3} + 2 (- 3 x^{2})$ .

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

Step 1.1.3.5.5

Factor $2$ out of $2 (x^{3} - 3 x^{2}) + 2 \cdot 4$ .

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 (x^{3} - 3 x^{2} + 4)}{{(x - 2)}^{2}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{2 (x^{3} - 3 x^{2} + 4)}{{(x - 2)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{2 (x^{3} - 3 x^{2} + 4)}{{(x - 2)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 (x^{3} - 3 x^{2} + 4) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $2 (x^{3} - 3 x^{2} + 4) = 0$ by $2$ and simplify.

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Step 2.3.1.1

Divide each term in $2 (x^{3} - 3 x^{2} + 4) = 0$ by $2$ .

$\frac{2 (x^{3} - 3 x^{2} + 4)}{2} = \frac{0}{2}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $2$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{2 (x^{3} - 3 x^{2} + 4)}{2} = \frac{0}{2}$

Step 2.3.1.2.1.2

Divide $x^{3} - 3 x^{2} + 4$ by $1$ .

$x^{3} - 3 x^{2} + 4 = \frac{0}{2}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $2$ .

$x^{3} - 3 x^{2} + 4 = 0$

Step 2.3.2

Factor the left side of the equation.

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Step 2.3.2.1

Factor $x^{3} - 3 x^{2} + 4$ using the rational roots test.

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Step 2.3.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1$

Step 2.3.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 4, \pm 2$

Step 2.3.2.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 2.3.2.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 3 {(- 1)}^{2} + 4$

Step 2.3.2.1.3.2

Raise $- 1$ to the power of $3$ .

$- 1 - 3 {(- 1)}^{2} + 4$

Step 2.3.2.1.3.3

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

$- 1 - 3 \cdot 1 + 4$

Step 2.3.2.1.3.4

Multiply $- 3$ by $1$ .

$- 1 - 3 + 4$

Step 2.3.2.1.3.5

Subtract $3$ from $- 1$ .

$- 4 + 4$

Step 2.3.2.1.3.6

Add $- 4$ and $4$ .

$0$

Step 2.3.2.1.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 3 x^{2} + 4}{x + 1}$

Step 2.3.2.1.5

Divide $x^{3} - 3 x^{2} + 4$ by $x + 1$ .

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Step 2.3.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{3}$

$3 x^{2}$

$0 x$

$4$

Step 2.3.2.1.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$

Step 2.3.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			+	$x^{3}$	+	$x^{2}$

Step 2.3.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + x^{2}$

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$

Step 2.3.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$

Step 2.3.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$

Step 2.3.2.1.5.7

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$

Step 2.3.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					-	$4 x^{2}$	-	$4 x$

Step 2.3.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} - 4 x$

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$

Step 2.3.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$

Step 2.3.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$	+	$4$

Step 2.3.2.1.5.12

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	+	$4$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$	+	$4$

Step 2.3.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$4$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$	+	$4$
							+	$4 x$	+	$4$

Step 2.3.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x + 4$

				$x^{2}$	-	$4 x$	+	$4$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$	+	$4$
							-	$4 x$	-	$4$

Step 2.3.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$4 x$	+	$4$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$4$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$0 x$
					+	$4 x^{2}$	+	$4 x$
							+	$4 x$	+	$4$
							-	$4 x$	-	$4$
										$0$

Step 2.3.2.1.5.16

Since the remander is $0$ , the final answer is the quotient.

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

Step 2.3.2.1.6

Write $x^{3} - 3 x^{2} + 4$ as a set of factors.

$(x + 1) (x^{2} - 4 x + 4) = 0$

Step 2.3.2.2

Factor using the perfect square rule.

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Step 2.3.2.2.1

Rewrite $4$ as $2^{2}$ .

$(x + 1) (x^{2} - 4 x + 2^{2}) = 0$

Step 2.3.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 2.3.2.2.3

Rewrite the polynomial.

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

Step 2.3.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 2.3.3

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

$x + 1 = 0$

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

Step 2.3.4

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

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Step 2.3.4.1

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

$x + 1 = 0$

Step 2.3.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.3.5

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.3.5.1

Set ${(x - 2)}^{2}$ equal to $0$ .

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

Step 2.3.5.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 2.3.5.2.1

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

$x - 2 = 0$

Step 2.3.5.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.3.6

The final solution is all the values that make $(x + 1) {(x - 2)}^{2} = 0$ true.

$x = - 1, 2$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{2 (x^{3} - 3 x^{2} + 4)}{{(x - 2)}^{2}}$ 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

Add $2$ to both sides of the equation.

$x = 2$

Step 4

Evaluate $\frac{x^{3} - 8}{x - 2}$ 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 $x$ .

$\frac{{(- 1)}^{3} - 8}{(- 1) - 2}$

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 $3$ .

$\frac{- 1 - 8}{- 1 - 2}$

Step 4.1.2.1.2

Subtract $8$ from $- 1$ .

$\frac{- 9}{- 1 - 2}$

Step 4.1.2.2

Simplify the expression.

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Step 4.1.2.2.1

Subtract $2$ from $- 1$ .

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

Step 4.1.2.2.2

Divide $- 9$ by $- 3$ .

$3$

Step 4.2

Evaluate at $x = 2$ .

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Step 4.2.1

Substitute $2$ for $x$ .

$\frac{{(2)}^{3} - 8}{(2) - 2}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Subtract $2$ from $2$ .

$\frac{{(2)}^{3} - 8}{0}$

Step 4.2.2.2

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

Undefined

Step 4.3

List all of the points.

$(- 1, 3)$

Step 5