Calculus Examples

\frac{2 x}{\sqrt{x^{2} - 9}}

$\frac{2 x}{\sqrt{x^{2} - 9}}$

Step 1

Find the

$y$ value at

$x = - 4$ .

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Step 1.1

Replace the variable

$x$ with

$- 4$ in the expression.

$f (- 4) = \frac{2 (- 4) \sqrt{((- 4) + 3) ((- 4) - 3)}}{((- 4) + 3) ((- 4) - 3)}$

Step 1.2

Simplify the result.

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Step 1.2.1

Multiply

$2$ by

$- 4$ .

$f (- 4) = \frac{- 8 \sqrt{(- 4 + 3) (- 4 - 3)}}{(- 4 + 3) (- 4 - 3)}$

Step 1.2.2

Simplify the denominator.

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Step 1.2.2.1

Add

$- 4$ and

$3$ .

$f (- 4) = \frac{- 8 \sqrt{(- 4 + 3) (- 4 - 3)}}{- 1 (- 4 - 3)}$

Step 1.2.2.2

Subtract

$3$ from

$- 4$ .

$f (- 4) = \frac{- 8 \sqrt{(- 4 + 3) (- 4 - 3)}}{- 1 \cdot - 7}$

Step 1.2.3

Simplify the numerator.

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Step 1.2.3.1

Add

$- 4$ and

$3$ .

$f (- 4) = \frac{- 8 \sqrt{- 1 (- 4 - 3)}}{- 1 \cdot - 7}$

Step 1.2.3.2

Subtract

$3$ from

$- 4$ .

$f (- 4) = \frac{- 8 \sqrt{- 1 \cdot - 7}}{- 1 \cdot - 7}$

Step 1.2.3.3

Multiply

$- 1$ by

$- 7$ .

$f (- 4) = \frac{- 8 \sqrt{7}}{- 1 \cdot - 7}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Multiply

$- 1$ by

$- 7$ .

$f (- 4) = \frac{- 8 \sqrt{7}}{7}$

Step 1.2.4.2

Move the negative in front of the fraction.

$f (- 4) = - \frac{8 \sqrt{7}}{7}$

Step 1.2.5

The final answer is

$- \frac{8 \sqrt{7}}{7}$ .

$- \frac{8 \sqrt{7}}{7}$

Step 1.3

The

$y$ value at

$x = - 4$ is

$- \frac{8 \sqrt{7}}{7}$ .

$y = - \frac{8 \sqrt{7}}{7}$

Step 2

Find the

$y$ value at

$x = - 5$ .

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Step 2.1

Replace the variable

$x$ with

$- 5$ in the expression.

$f (- 5) = \frac{2 (- 5) \sqrt{((- 5) + 3) ((- 5) - 3)}}{((- 5) + 3) ((- 5) - 3)}$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply

$2$ by

$- 5$ .

$f (- 5) = \frac{- 10 \sqrt{(- 5 + 3) (- 5 - 3)}}{(- 5 + 3) (- 5 - 3)}$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Add

$- 5$ and

$3$ .

$f (- 5) = \frac{- 10 \sqrt{(- 5 + 3) (- 5 - 3)}}{- 2 (- 5 - 3)}$

Step 2.2.2.2

Subtract

$3$ from

$- 5$ .

$f (- 5) = \frac{- 10 \sqrt{(- 5 + 3) (- 5 - 3)}}{- 2 \cdot - 8}$

Step 2.2.3

Simplify the numerator.

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Step 2.2.3.1

Add

$- 5$ and

$3$ .

$f (- 5) = \frac{- 10 \sqrt{- 2 (- 5 - 3)}}{- 2 \cdot - 8}$

Step 2.2.3.2

Subtract

$3$ from

$- 5$ .

$f (- 5) = \frac{- 10 \sqrt{- 2 \cdot - 8}}{- 2 \cdot - 8}$

Step 2.2.3.3

Multiply

$- 2$ by

$- 8$ .

$f (- 5) = \frac{- 10 \sqrt{16}}{- 2 \cdot - 8}$

Step 2.2.3.4

Rewrite

$16$ as

$4^{2}$ .

$f (- 5) = \frac{- 10 \sqrt{4^{2}}}{- 2 \cdot - 8}$

Step 2.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$f (- 5) = \frac{- 10 \cdot 4}{- 2 \cdot - 8}$

Step 2.2.4

Reduce the expression by cancelling the common factors.

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Step 2.2.4.1

Multiply

$- 2$ by

$- 8$ .

$f (- 5) = \frac{- 10 \cdot 4}{16}$

Step 2.2.4.2

Multiply

$- 10$ by

$4$ .

$f (- 5) = \frac{- 40}{16}$

Step 2.2.4.3

Cancel the common factor of

$- 40$ and

$16$ .

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Step 2.2.4.3.1

Factor

$8$ out of

$- 40$ .

$f (- 5) = \frac{8 (- 5)}{16}$

Step 2.2.4.3.2

Cancel the common factors.

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Step 2.2.4.3.2.1

Factor

$8$ out of

$16$ .

$f (- 5) = \frac{8 \cdot - 5}{8 \cdot 2}$

Step 2.2.4.3.2.2

Cancel the common factor.

$f (- 5) = \frac{8 \cdot - 5}{8 \cdot 2}$

Step 2.2.4.3.2.3

Rewrite the expression.

$f (- 5) = \frac{- 5}{2}$

Step 2.2.4.4

Move the negative in front of the fraction.

$f (- 5) = - \frac{5}{2}$

Step 2.2.5

The final answer is

$- \frac{5}{2}$ .

$- \frac{5}{2}$

Step 2.3

The

$y$ value at

$x = - 5$ is

$- \frac{5}{2}$ .

$y = - \frac{5}{2}$

Step 3

Find the

$y$ value at

$x = - 6$ .

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Step 3.1

Replace the variable

$x$ with

$- 6$ in the expression.

$f (- 6) = \frac{2 (- 6) \sqrt{((- 6) + 3) ((- 6) - 3)}}{((- 6) + 3) ((- 6) - 3)}$

Step 3.2

Simplify the result.

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Step 3.2.1

Reduce the expression by cancelling the common factors.

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Step 3.2.1.1

Cancel the common factor of

$- 6$ and

$(- 6) + 3$ .

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Step 3.2.1.1.1

Factor

$3$ out of

$2 (- 6) \sqrt{((- 6) + 3) ((- 6) - 3)}$ .

$f (- 6) = \frac{3 (2 \cdot ((- 2) \sqrt{((- 6) + 3) ((- 6) - 3)}))}{((- 6) + 3) ((- 6) - 3)}$

Step 3.2.1.1.2

Cancel the common factors.

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Step 3.2.1.1.2.1

Factor

$3$ out of

$((- 6) + 3) ((- 6) - 3)$ .

$f (- 6) = \frac{3 (2 \cdot ((- 2) \sqrt{((- 6) + 3) ((- 6) - 3)}))}{3 ((- 2 + 1) ((- 6) - 3))}$

Step 3.2.1.1.2.2

Cancel the common factor.

$f (- 6) = \frac{3 (2 \cdot ((- 2) \sqrt{((- 6) + 3) ((- 6) - 3)}))}{3 ((- 2 + 1) ((- 6) - 3))}$

Step 3.2.1.1.2.3

Rewrite the expression.

$f (- 6) = \frac{2 \cdot ((- 2) \sqrt{((- 6) + 3) ((- 6) - 3)})}{(- 2 + 1) ((- 6) - 3)}$

Step 3.2.1.2

Multiply

$2$ by

$- 2$ .

$f (- 6) = \frac{- 4 \sqrt{(- 6 + 3) (- 6 - 3)}}{(- 2 + 1) (- 6 - 3)}$

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

Add

$- 2$ and

$1$ .

$f (- 6) = \frac{- 4 \sqrt{(- 6 + 3) (- 6 - 3)}}{- 1 (- 6 - 3)}$

Step 3.2.2.2

Subtract

$3$ from

$- 6$ .

$f (- 6) = \frac{- 4 \sqrt{(- 6 + 3) (- 6 - 3)}}{- 1 \cdot - 9}$

Step 3.2.3

Simplify the numerator.

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Step 3.2.3.1

Add

$- 6$ and

$3$ .

$f (- 6) = \frac{- 4 \sqrt{- 3 (- 6 - 3)}}{- 1 \cdot - 9}$

Step 3.2.3.2

Subtract

$3$ from

$- 6$ .

$f (- 6) = \frac{- 4 \sqrt{- 3 \cdot - 9}}{- 1 \cdot - 9}$

Step 3.2.3.3

Multiply

$- 3$ by

$- 9$ .

$f (- 6) = \frac{- 4 \sqrt{27}}{- 1 \cdot - 9}$

Step 3.2.3.4

Rewrite

$27$ as

$3^{2} \cdot 3$ .

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Step 3.2.3.4.1

Factor

$9$ out of

$27$ .

$f (- 6) = \frac{- 4 \sqrt{9 (3)}}{- 1 \cdot - 9}$

Step 3.2.3.4.2

Rewrite

$9$ as

$3^{2}$ .

$f (- 6) = \frac{- 4 \sqrt{3^{2} \cdot 3}}{- 1 \cdot - 9}$

Step 3.2.3.5

Pull terms out from under the radical.

$f (- 6) = \frac{- 4 \cdot (3 \sqrt{3})}{- 1 \cdot - 9}$

Step 3.2.3.6

Multiply

$- 4$ by

$3$ .

$f (- 6) = \frac{- 12 \sqrt{3}}{- 1 \cdot - 9}$

Step 3.2.4

Reduce the expression by cancelling the common factors.

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Step 3.2.4.1

Multiply

$- 1$ by

$- 9$ .

$f (- 6) = \frac{- 12 \sqrt{3}}{9}$

Step 3.2.4.2

Cancel the common factor of

$- 12$ and

$9$ .

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Step 3.2.4.2.1

Factor

$3$ out of

$- 12 \sqrt{3}$ .

$f (- 6) = \frac{3 (- 4 \sqrt{3})}{9}$

Step 3.2.4.2.2

Cancel the common factors.

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Step 3.2.4.2.2.1

Factor

$3$ out of

$9$ .

$f (- 6) = \frac{3 (- 4 \sqrt{3})}{3 (3)}$

Step 3.2.4.2.2.2

Cancel the common factor.

$f (- 6) = \frac{3 (- 4 \sqrt{3})}{3 \cdot 3}$

Step 3.2.4.2.2.3

Rewrite the expression.

$f (- 6) = \frac{- 4 \sqrt{3}}{3}$

Step 3.2.4.3

Move the negative in front of the fraction.

$f (- 6) = - \frac{4 \sqrt{3}}{3}$

Step 3.2.5

The final answer is

$- \frac{4 \sqrt{3}}{3}$ .

$- \frac{4 \sqrt{3}}{3}$

Step 3.3

The

$y$ value at

$x = - 6$ is

$- \frac{4 \sqrt{3}}{3}$ .

$y = - \frac{4 \sqrt{3}}{3}$

Step 4

Find the

$y$ value at

$x = 4$ .

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Step 4.1

Replace the variable

$x$ with

$4$ in the expression.

$f (4) = \frac{2 (4) \sqrt{((4) + 3) ((4) - 3)}}{((4) + 3) ((4) - 3)}$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply

$2$ by

$4$ .

$f (4) = \frac{8 \sqrt{(4 + 3) (4 - 3)}}{(4 + 3) (4 - 3)}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

Add

$4$ and

$3$ .

$f (4) = \frac{8 \sqrt{(4 + 3) (4 - 3)}}{7 (4 - 3)}$

Step 4.2.2.2

Subtract

$3$ from

$4$ .

$f (4) = \frac{8 \sqrt{(4 + 3) (4 - 3)}}{7 \cdot 1}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Add

$4$ and

$3$ .

$f (4) = \frac{8 \sqrt{7 (4 - 3)}}{7 \cdot 1}$

Step 4.2.3.2

Subtract

$3$ from

$4$ .

$f (4) = \frac{8 \sqrt{7 \cdot 1}}{7 \cdot 1}$

Step 4.2.3.3

Multiply

$7$ by

$1$ .

$f (4) = \frac{8 \sqrt{7}}{7 \cdot 1}$

Step 4.2.4

Multiply

$7$ by

$1$ .

$f (4) = \frac{8 \sqrt{7}}{7}$

Step 4.2.5

The final answer is

$\frac{8 \sqrt{7}}{7}$ .

$\frac{8 \sqrt{7}}{7}$

Step 4.3

The

$y$ value at

$x = 4$ is

$\frac{8 \sqrt{7}}{7}$ .

$y = \frac{8 \sqrt{7}}{7}$

Step 5

Find the

$y$ value at

$x = 5$ .

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Step 5.1

Replace the variable

$x$ with

$5$ in the expression.

$f (5) = \frac{2 (5) \sqrt{((5) + 3) ((5) - 3)}}{((5) + 3) ((5) - 3)}$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply

$2$ by

$5$ .

$f (5) = \frac{10 \sqrt{(5 + 3) (5 - 3)}}{(5 + 3) (5 - 3)}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Add

$5$ and

$3$ .

$f (5) = \frac{10 \sqrt{(5 + 3) (5 - 3)}}{8 (5 - 3)}$

Step 5.2.2.2

Subtract

$3$ from

$5$ .

$f (5) = \frac{10 \sqrt{(5 + 3) (5 - 3)}}{8 \cdot 2}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Add

$5$ and

$3$ .

$f (5) = \frac{10 \sqrt{8 (5 - 3)}}{8 \cdot 2}$

Step 5.2.3.2

Subtract

$3$ from

$5$ .

$f (5) = \frac{10 \sqrt{8 \cdot 2}}{8 \cdot 2}$

Step 5.2.3.3

Multiply

$8$ by

$2$ .

$f (5) = \frac{10 \sqrt{16}}{8 \cdot 2}$

Step 5.2.3.4

Rewrite

$16$ as

$4^{2}$ .

$f (5) = \frac{10 \sqrt{4^{2}}}{8 \cdot 2}$

Step 5.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$f (5) = \frac{10 \cdot 4}{8 \cdot 2}$

Step 5.2.4

Reduce the expression by cancelling the common factors.

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Step 5.2.4.1

Multiply

$8$ by

$2$ .

$f (5) = \frac{10 \cdot 4}{16}$

Step 5.2.4.2

Multiply

$10$ by

$4$ .

$f (5) = \frac{40}{16}$

Step 5.2.4.3

Cancel the common factor of

$40$ and

$16$ .

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Step 5.2.4.3.1

Factor

$8$ out of

$40$ .

$f (5) = \frac{8 (5)}{16}$

Step 5.2.4.3.2

Cancel the common factors.

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Step 5.2.4.3.2.1

Factor

$8$ out of

$16$ .

$f (5) = \frac{8 \cdot 5}{8 \cdot 2}$

Step 5.2.4.3.2.2

Cancel the common factor.

$f (5) = \frac{8 \cdot 5}{8 \cdot 2}$

Step 5.2.4.3.2.3

Rewrite the expression.

$f (5) = \frac{5}{2}$

Step 5.2.5

The final answer is

$\frac{5}{2}$ .

$\frac{5}{2}$

Step 5.3

The

$y$ value at

$x = 5$ is

$\frac{5}{2}$ .

$y = \frac{5}{2}$

Step 6

Find the

$y$ value at

$x = 6$ .

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Step 6.1

Replace the variable

$x$ with

$6$ in the expression.

$f (6) = \frac{2 (6) \sqrt{((6) + 3) ((6) - 3)}}{((6) + 3) ((6) - 3)}$

Step 6.2

Simplify the result.

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Step 6.2.1

Reduce the expression by cancelling the common factors.

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Step 6.2.1.1

Cancel the common factor of

$6$ and

$(6) + 3$ .

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Step 6.2.1.1.1

Factor

$3$ out of

$2 (6) \sqrt{((6) + 3) ((6) - 3)}$ .

$f (6) = \frac{3 (2 \cdot ((2) \sqrt{((6) + 3) ((6) - 3)}))}{((6) + 3) ((6) - 3)}$

Step 6.2.1.1.2

Cancel the common factors.

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Step 6.2.1.1.2.1

Factor

$3$ out of

$((6) + 3) ((6) - 3)$ .

$f (6) = \frac{3 (2 \cdot ((2) \sqrt{((6) + 3) ((6) - 3)}))}{3 ((2 + 1) ((6) - 3))}$

Step 6.2.1.1.2.2

Cancel the common factor.

$f (6) = \frac{3 (2 \cdot ((2) \sqrt{((6) + 3) ((6) - 3)}))}{3 ((2 + 1) ((6) - 3))}$

Step 6.2.1.1.2.3

Rewrite the expression.

$f (6) = \frac{2 \cdot ((2) \sqrt{((6) + 3) ((6) - 3)})}{(2 + 1) ((6) - 3)}$

Step 6.2.1.2

Multiply

$2$ by

$2$ .

$f (6) = \frac{4 \sqrt{(6 + 3) (6 - 3)}}{(2 + 1) (6 - 3)}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add

$2$ and

$1$ .

$f (6) = \frac{4 \sqrt{(6 + 3) (6 - 3)}}{3 (6 - 3)}$

Step 6.2.2.2

Subtract

$3$ from

$6$ .

$f (6) = \frac{4 \sqrt{(6 + 3) (6 - 3)}}{3 \cdot 3}$

Step 6.2.3

Simplify the numerator.

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Step 6.2.3.1

Add

$6$ and

$3$ .

$f (6) = \frac{4 \sqrt{9 (6 - 3)}}{3 \cdot 3}$

Step 6.2.3.2

Subtract

$3$ from

$6$ .

$f (6) = \frac{4 \sqrt{9 \cdot 3}}{3 \cdot 3}$

Step 6.2.3.3

Multiply

$9$ by

$3$ .

$f (6) = \frac{4 \sqrt{27}}{3 \cdot 3}$

Step 6.2.3.4

Rewrite

$27$ as

$3^{2} \cdot 3$ .

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Step 6.2.3.4.1

Factor

$9$ out of

$27$ .

$f (6) = \frac{4 \sqrt{9 (3)}}{3 \cdot 3}$

Step 6.2.3.4.2

Rewrite

$9$ as

$3^{2}$ .

$f (6) = \frac{4 \sqrt{3^{2} \cdot 3}}{3 \cdot 3}$

Step 6.2.3.5

Pull terms out from under the radical.

$f (6) = \frac{4 \cdot (3 \sqrt{3})}{3 \cdot 3}$

Step 6.2.3.6

Multiply

$4$ by

$3$ .

$f (6) = \frac{12 \sqrt{3}}{3 \cdot 3}$

Step 6.2.4

Reduce the expression by cancelling the common factors.

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Step 6.2.4.1

Multiply

$3$ by

$3$ .

$f (6) = \frac{12 \sqrt{3}}{9}$

Step 6.2.4.2

Cancel the common factor of

$12$ and

$9$ .

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Step 6.2.4.2.1

Factor

$3$ out of

$12 \sqrt{3}$ .

$f (6) = \frac{3 (4 \sqrt{3})}{9}$

Step 6.2.4.2.2

Cancel the common factors.

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Step 6.2.4.2.2.1

Factor

$3$ out of

$9$ .

$f (6) = \frac{3 (4 \sqrt{3})}{3 (3)}$

Step 6.2.4.2.2.2

Cancel the common factor.

$f (6) = \frac{3 (4 \sqrt{3})}{3 \cdot 3}$

Step 6.2.4.2.2.3

Rewrite the expression.

$f (6) = \frac{4 \sqrt{3}}{3}$

Step 6.2.5

The final answer is

$\frac{4 \sqrt{3}}{3}$ .

$\frac{4 \sqrt{3}}{3}$

Step 6.3

The

$y$ value at

$x = 6$ is

$\frac{4 \sqrt{3}}{3}$ .

$y = \frac{4 \sqrt{3}}{3}$

Step 7

List the points to graph.

$(- 4, - 3.02371578), (- 5, - 2.5), (- 6, - 2.30940107), (4, 3.02371578), (5, 2.5), (6, 2.30940107)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 6 & - 2.309 \\ - 5 & - 2.5 \\ - 4 & - 3.024 \\ 4 & 3.024 \\ 5 & 2.5 \\ 6 & 2.309 \end{array}$

Step 9