Calculus Examples

$f (x) = \sqrt{x} (x - 8)$ , $(16, 32)$

Step 1

Find the first derivative and evaluate at $x = 16$ and $y = 32$ to find the slope of the tangent line.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - 8$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

$x^{\frac{1}{2}} (1 + 0) + (x - 8) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 1.3.4

Simplify the expression.

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Step 1.3.4.1

Add $1$ and $0$ .

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

Step 1.3.4.2

Multiply $x^{\frac{1}{2}}$ by $1$ .

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

Step 1.3.5

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

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

Step 1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Move the negative in front of the fraction.

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

Step 1.9

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

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

Step 1.10

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.11

Simplify.

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Step 1.11.1

Apply the distributive property.

$x^{\frac{1}{2}} + x \frac{1}{2 x^{\frac{1}{2}}} - 8 \frac{1}{2 x^{\frac{1}{2}}}$

Step 1.11.2

Combine terms.

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Step 1.11.2.1

Combine $x$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$x^{\frac{1}{2}} + \frac{x}{2 x^{\frac{1}{2}}} - 8 \frac{1}{2 x^{\frac{1}{2}}}$

Step 1.11.2.2

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 1.11.2.3

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

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Step 1.11.2.3.1

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

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Step 1.11.2.3.1.1

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

$x^{\frac{1}{2}} + \frac{x^{1} x^{- \frac{1}{2}}}{2} - 8 \frac{1}{2 x^{\frac{1}{2}}}$

Step 1.11.2.3.1.2

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

$x^{\frac{1}{2}} + \frac{x^{1 - \frac{1}{2}}}{2} - 8 \frac{1}{2 x^{\frac{1}{2}}}$

Step 1.11.2.3.2

Write $1$ as a fraction with a common denominator.

$x^{\frac{1}{2}} + \frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} - 8 \frac{1}{2 x^{\frac{1}{2}}}$

Step 1.11.2.3.3

Combine the numerators over the common denominator.

$x^{\frac{1}{2}} + \frac{x^{\frac{2 - 1}{2}}}{2} - 8 \frac{1}{2 x^{\frac{1}{2}}}$

Step 1.11.2.3.4

Subtract $1$ from $2$ .

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} - 8 \frac{1}{2 x^{\frac{1}{2}}}$

Step 1.11.2.4

Combine $- 8$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{- 8}{2 x^{\frac{1}{2}}}$

Step 1.11.2.5

Factor $2$ out of $- 8$ .

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

Step 1.11.2.6

Cancel the common factors.

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Step 1.11.2.6.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 1.11.2.6.2

Cancel the common factor.

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

Step 1.11.2.6.3

Rewrite the expression.

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

Step 1.11.2.7

Move the negative in front of the fraction.

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

Step 1.11.2.8

To write $x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.11.2.9

Combine $x^{\frac{1}{2}}$ and $\frac{2}{2}$ .

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

Step 1.11.2.10

Combine the numerators over the common denominator.

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

Step 1.11.2.11

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

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

Step 1.11.2.12

Add $2 x^{\frac{1}{2}}$ and $x^{\frac{1}{2}}$ .

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

Step 1.12

Evaluate the derivative at $x = 16$ .

$\frac{3 {(16)}^{\frac{1}{2}}}{2} - \frac{4}{{(16)}^{\frac{1}{2}}}$

Step 1.13

Simplify.

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Step 1.13.1

Simplify each term.

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Step 1.13.1.1

Simplify the numerator.

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Step 1.13.1.1.1

Rewrite $16$ as $4^{2}$ .

$\frac{3 \cdot {(4^{2})}^{\frac{1}{2}}}{2} - \frac{4}{{(16)}^{\frac{1}{2}}}$

Step 1.13.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 \cdot 4^{2 (\frac{1}{2})}}{2} - \frac{4}{{(16)}^{\frac{1}{2}}}$

Step 1.13.1.1.3

Cancel the common factor of $2$ .

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Step 1.13.1.1.3.1

Cancel the common factor.

$\frac{3 \cdot 4^{2 (\frac{1}{2})}}{2} - \frac{4}{{(16)}^{\frac{1}{2}}}$

Step 1.13.1.1.3.2

Rewrite the expression.

$\frac{3 \cdot 4^{1}}{2} - \frac{4}{{(16)}^{\frac{1}{2}}}$

Step 1.13.1.1.4

Evaluate the exponent.

$\frac{3 \cdot 4}{2} - \frac{4}{{(16)}^{\frac{1}{2}}}$

Step 1.13.1.2

Multiply $3$ by $4$ .

$\frac{12}{2} - \frac{4}{{(16)}^{\frac{1}{2}}}$

Step 1.13.1.3

Divide $12$ by $2$ .

$6 - \frac{4}{{(16)}^{\frac{1}{2}}}$

Step 1.13.1.4

Simplify the denominator.

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Step 1.13.1.4.1

Rewrite $16$ as $4^{2}$ .

$6 - \frac{4}{{(4^{2})}^{\frac{1}{2}}}$

Step 1.13.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$6 - \frac{4}{4^{2 (\frac{1}{2})}}$

Step 1.13.1.4.3

Cancel the common factor of $2$ .

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Step 1.13.1.4.3.1

Cancel the common factor.

$6 - \frac{4}{4^{2 (\frac{1}{2})}}$

Step 1.13.1.4.3.2

Rewrite the expression.

$6 - \frac{4}{4^{1}}$

Step 1.13.1.4.4

Evaluate the exponent.

$6 - \frac{4}{4}$

Step 1.13.1.5

Cancel the common factor of $4$ .

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Step 1.13.1.5.1

Cancel the common factor.

$6 - \frac{4}{4}$

Step 1.13.1.5.2

Rewrite the expression.

$6 - 1 \cdot 1$

Step 1.13.1.6

Multiply $- 1$ by $1$ .

$6 - 1$

Step 1.13.2

Subtract $1$ from $6$ .

$5$

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 $5$ and a given point $(16, 32)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (32) = 5 \cdot (x - (16))$

Step 2.2

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

$y - 32 = 5 \cdot (x - 16)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $5 \cdot (x - 16)$ .

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Step 2.3.1.1

Rewrite.

$y - 32 = 0 + 0 + 5 \cdot (x - 16)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 32 = 5 \cdot (x - 16)$

Step 2.3.1.3

Apply the distributive property.

$y - 32 = 5 x + 5 \cdot - 16$

Step 2.3.1.4

Multiply $5$ by $- 16$ .

$y - 32 = 5 x - 80$

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 $32$ to both sides of the equation.

$y = 5 x - 80 + 32$

Step 2.3.2.2

Add $- 80$ and $32$ .

$y = 5 x - 48$

Step 3