Calculus Examples

y = \sqrt{2 x}

$y = \sqrt{2 x}$ at

(8, 4)

$(8, 4)$

Step 1

Find the first derivative and evaluate at

x = 8

$x = 8$ and

y = 4

$y = 4$ to find the slope of the tangent line.

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Step 1.1

Simplify with factoring out.

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Step 1.1.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2 x}

$\sqrt{2 x}$ as

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

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

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

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

Step 1.1.2

Factor

2

$2$ out of

2 x

$2 x$ .

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

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

Step 1.1.3

Apply the product rule to

2 (x)

$2 (x)$ .

\frac{d}{d x} [2^{\frac{1}{2}} x^{\frac{1}{2}}]

$\frac{d}{d x} [2^{\frac{1}{2}} x^{\frac{1}{2}}]$

\frac{d}{d x} [2^{\frac{1}{2}} x^{\frac{1}{2}}]

$\frac{d}{d x} [2^{\frac{1}{2}} x^{\frac{1}{2}}]$

Step 1.2

Since

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ is constant with respect to

x

$x$ , the derivative of

2^{\frac{1}{2}} x^{\frac{1}{2}}

$2^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to

x

$x$ is

2^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]

$2^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

2^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]

$2^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 1.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{1}{2}

$n = \frac{1}{2}$ .

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

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

Step 1.4

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.5

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.6

Combine the numerators over the common denominator.

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

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 1.7.2

Subtract

2

$2$ from

1

$1$ .

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

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

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

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

Step 1.8

Move the negative in front of the fraction.

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

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

Step 1.9

Combine

\frac{1}{2}

$\frac{1}{2}$ and

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

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

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

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

Step 1.10

Combine

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ and

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

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

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

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

Step 1.11

Simplify the expression.

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Step 1.11.1

Move

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ to the denominator using the negative exponent rule

b^{n} = \frac{1}{b^{- n}}

$b^{n} = \frac{1}{b^{- n}}$ .

\frac{x^{- \frac{1}{2}}}{2 \cdot 2^{- \frac{1}{2}}}

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

Step 1.11.2

Move

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

$x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

\frac{1}{2 \cdot 2^{- \frac{1}{2}} x^{\frac{1}{2}}}

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

\frac{1}{2 \cdot 2^{- \frac{1}{2}} x^{\frac{1}{2}}}

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

Step 1.12

Multiply

2

$2$ by

2^{- \frac{1}{2}}

$2^{- \frac{1}{2}}$ by adding the exponents.

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Step 1.12.1

Multiply

2

$2$ by

2^{- \frac{1}{2}}

$2^{- \frac{1}{2}}$ .

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Step 1.12.1.1

Raise

2

$2$ to the power of

1

$1$ .

\frac{1}{2^{1} \cdot 2^{- \frac{1}{2}} x^{\frac{1}{2}}}

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

Step 1.12.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 1.12.2

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 1.12.3

Combine the numerators over the common denominator.

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

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

Step 1.12.4

Subtract

1

$1$ from

2

$2$ .

\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}

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

\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}

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

Step 1.13

Evaluate the derivative at

x = 8

$x = 8$ .

\frac{1}{2^{\frac{1}{2}} {(8)}^{\frac{1}{2}}}

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

Step 1.14

Simplify.

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Step 1.14.1

Simplify the denominator.

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Step 1.14.1.1

Rewrite

8

$8$ as

2^{3}

$2^{3}$ .

\frac{1}{2^{\frac{1}{2}} \cdot {(2^{3})}^{\frac{1}{2}}}

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

Step 1.14.1.2

Multiply the exponents in

{(2^{3})}^{\frac{1}{2}}

${(2^{3})}^{\frac{1}{2}}$ .

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Step 1.14.1.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{1}{2^{\frac{1}{2}} \cdot 2^{3 (\frac{1}{2})}}

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

Step 1.14.1.2.2

Combine

3

$3$ and

\frac{1}{2}

$\frac{1}{2}$ .

\frac{1}{2^{\frac{1}{2}} \cdot 2^{\frac{3}{2}}}

$\frac{1}{2^{\frac{1}{2}} \cdot 2^{\frac{3}{2}}}$

\frac{1}{2^{\frac{1}{2}} \cdot 2^{\frac{3}{2}}}

$\frac{1}{2^{\frac{1}{2}} \cdot 2^{\frac{3}{2}}}$

Step 1.14.1.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{1}{2^{\frac{1}{2} + \frac{3}{2}}}

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

Step 1.14.1.4

Combine the numerators over the common denominator.

\frac{1}{2^{\frac{1 + 3}{2}}}

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

Step 1.14.1.5

Add

1

$1$ and

3

$3$ .

\frac{1}{2^{\frac{4}{2}}}

$\frac{1}{2^{\frac{4}{2}}}$

Step 1.14.1.6

Cancel the common factor of

4

$4$ and

2

$2$ .

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Step 1.14.1.6.1

Factor

2

$2$ out of

4

$4$ .

\frac{1}{2^{\frac{2 \cdot 2}{2}}}

$\frac{1}{2^{\frac{2 \cdot 2}{2}}}$

Step 1.14.1.6.2

Cancel the common factors.

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Step 1.14.1.6.2.1

Factor

2

$2$ out of

2

$2$ .

\frac{1}{2^{\frac{2 \cdot 2}{2 (1)}}}

$\frac{1}{2^{\frac{2 \cdot 2}{2 (1)}}}$

Step 1.14.1.6.2.2

Cancel the common factor.

$\frac{1}{2^{\frac{2 \cdot 2}{2 \cdot 1}}}$

Step 1.14.1.6.2.3

Rewrite the expression.

$\frac{1}{2^{\frac{2}{1}}}$

Step 1.14.1.6.2.4

Divide

$2$ by

$1$ .

$\frac{1}{2^{2}}$

Step 1.14.2

Raise

$2$ to the power of

$2$ .

$\frac{1}{4}$

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

$\frac{1}{4}$ and a given point

$(8, 4)$ 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

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = \frac{1}{4} \cdot (x - (8))$

Step 2.2

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

$y - 4 = \frac{1}{4} \cdot (x - 8)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$\frac{1}{4} \cdot (x - 8)$ .

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Step 2.3.1.1

Rewrite.

$y - 4 = 0 + 0 + \frac{1}{4} \cdot (x - 8)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 = \frac{1}{4} \cdot (x - 8)$

Step 2.3.1.3

Apply the distributive property.

$y - 4 = \frac{1}{4} x + \frac{1}{4} \cdot - 8$

Step 2.3.1.4

Combine

$\frac{1}{4}$ and

$x$ .

$y - 4 = \frac{x}{4} + \frac{1}{4} \cdot - 8$

Step 2.3.1.5

Cancel the common factor of

$4$ .

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Step 2.3.1.5.1

Factor

$4$ out of

$- 8$ .

$y - 4 = \frac{x}{4} + \frac{1}{4} \cdot (4 (- 2))$

Step 2.3.1.5.2

Cancel the common factor.

$y - 4 = \frac{x}{4} + \frac{1}{4} \cdot (4 \cdot - 2)$

Step 2.3.1.5.3

Rewrite the expression.

$y - 4 = \frac{x}{4} - 2$

Step 2.3.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 2.3.2.1

Add

$4$ to both sides of the equation.

$y = \frac{x}{4} - 2 + 4$

Step 2.3.2.2

Add

$- 2$ and

$4$ .

$y = \frac{x}{4} + 2$

Step 2.3.3

Reorder terms.

$y = \frac{1}{4} x + 2$

Step 3