Calculus Examples

$f (x) = \sqrt{x}$ ; $x = 4$

Step 1

Find the corresponding $y$ -value to $x = 4$ .

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Step 1.1

Substitute $4$ in for $x$ .

$y = \sqrt{4}$

Step 1.2

Simplify $\sqrt{4}$ .

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Step 1.2.1

Remove parentheses.

$y = \sqrt{4}$

Step 1.2.2

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

$y = \sqrt{2^{2}}$

Step 1.2.3

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

$y = 2$

Step 2

Find the first derivative and evaluate at $x = 4$ and $y = 2$ to find the slope of the tangent line.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $1$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 2.8

Simplify.

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Step 2.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8.2

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

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

Step 2.9

Evaluate the derivative at $x = 4$ .

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

Step 2.10

Simplify.

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Step 2.10.1

Simplify the denominator.

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Step 2.10.1.1

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

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

Step 2.10.1.2

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

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Step 2.10.1.2.1

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

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

Step 2.10.1.2.2

Cancel the common factor of $2$ .

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Step 2.10.1.2.2.1

Cancel the common factor.

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

Step 2.10.1.2.2.2

Rewrite the expression.

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

Step 2.10.1.3

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

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

Step 2.10.1.4

Add $1$ and $1$ .

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

Step 2.10.2

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

$\frac{1}{4}$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $\frac{1}{4}$ and a given point $(4, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite.

$y - 2 = 0 + 0 + \frac{1}{4} \cdot (x - 4)$

Step 3.3.1.2

Simplify by adding zeros.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.1.4

Combine $\frac{1}{4}$ and $x$ .

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

Step 3.3.1.5

Cancel the common factor of $4$ .

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Step 3.3.1.5.1

Factor $4$ out of $- 4$ .

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

Step 3.3.1.5.2

Cancel the common factor.

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

Step 3.3.1.5.3

Rewrite the expression.

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

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $2$ to both sides of the equation.

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

Step 3.3.2.2

Add $- 1$ and $2$ .

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

Step 3.3.3

Reorder terms.

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

Step 4