Calculus Examples

$f (x) = \sqrt{2 x}$ at $(32, 8)$

Step 1

Find the first derivative and evaluate at $x = 32$ and $y = 8$ 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 $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x}$ as ${(2 x)}^{\frac{1}{2}}$ .

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

Step 1.1.2

Factor $2$ out of $2 x$ .

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

Step 1.1.3

Apply the product rule to $2 (x)$ .

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

Step 1.2

Since $2^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $2^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $2^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Move the negative in front of the fraction.

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

Simplify the expression.

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Step 1.11.1

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

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

Step 1.11.2

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

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

Step 1.12

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

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Step 1.12.1

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

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Step 1.12.1.1

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

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

Step 1.12.1.2

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

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

Step 1.12.2

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

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

Step 1.12.3

Combine the numerators over the common denominator.

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

Step 1.12.4

Subtract $1$ from $2$ .

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

Step 1.13

Evaluate the derivative at $x = 32$ .

$\frac{1}{2^{\frac{1}{2}} {(32)}^{\frac{1}{2}}}$

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 $32$ as $2^{5}$ .

$\frac{1}{2^{\frac{1}{2}} \cdot {(2^{5})}^{\frac{1}{2}}}$

Step 1.14.1.2

Multiply the exponents in ${(2^{5})}^{\frac{1}{2}}$ .

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Step 1.14.1.2.1

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

$\frac{1}{2^{\frac{1}{2}} \cdot 2^{5 (\frac{1}{2})}}$

Step 1.14.1.2.2

Combine $5$ and $\frac{1}{2}$ .

$\frac{1}{2^{\frac{1}{2}} \cdot 2^{\frac{5}{2}}}$

Step 1.14.1.3

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

$\frac{1}{2^{\frac{1}{2} + \frac{5}{2}}}$

Step 1.14.1.4

Combine the numerators over the common denominator.

$\frac{1}{2^{\frac{1 + 5}{2}}}$

Step 1.14.1.5

Add $1$ and $5$ .

$\frac{1}{2^{\frac{6}{2}}}$

Step 1.14.1.6

Cancel the common factor of $6$ and $2$ .

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Step 1.14.1.6.1

Factor $2$ out of $6$ .

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

Step 1.14.1.6.2

Cancel the common factors.

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Step 1.14.1.6.2.1

Factor $2$ out of $2$ .

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

Step 1.14.1.6.2.2

Cancel the common factor.

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

Step 1.14.1.6.2.3

Rewrite the expression.

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

Step 1.14.1.6.2.4

Divide $3$ by $1$ .

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

Step 1.14.2

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

$\frac{1}{8}$

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 $\frac{1}{8}$ and a given point $(32, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = \frac{1}{8} \cdot (x - (32))$

Step 2.2

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

$y - 8 = \frac{1}{8} \cdot (x - 32)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{8} \cdot (x - 32)$ .

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Step 2.3.1.1

Rewrite.

$y - 8 = 0 + 0 + \frac{1}{8} \cdot (x - 32)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 8 = \frac{1}{8} \cdot (x - 32)$

Step 2.3.1.3

Apply the distributive property.

$y - 8 = \frac{1}{8} x + \frac{1}{8} \cdot - 32$

Step 2.3.1.4

Combine $\frac{1}{8}$ and $x$ .

$y - 8 = \frac{x}{8} + \frac{1}{8} \cdot - 32$

Step 2.3.1.5

Cancel the common factor of $8$ .

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Step 2.3.1.5.1

Factor $8$ out of $- 32$ .

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

Step 2.3.1.5.2

Cancel the common factor.

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

Step 2.3.1.5.3

Rewrite the expression.

$y - 8 = \frac{x}{8} - 4$

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 $8$ to both sides of the equation.

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

Step 2.3.2.2

Add $- 4$ and $8$ .

$y = \frac{x}{8} + 4$

Step 2.3.3

Reorder terms.

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

Step 3