Calculus Examples

$y = 2 \sqrt{x}$ , $(16, 8)$

Step 1

Find the first derivative and evaluate at $x = 16$ and $y = 8$ 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 $x^{\frac{1}{2}}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Move the negative in front of the fraction.

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

Cancel the common factor.

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

Step 1.13

Rewrite the expression.

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

Step 1.14

Evaluate the derivative at $x = 16$ .

$\frac{1}{{(16)}^{\frac{1}{2}}}$

Step 1.15

Simplify the denominator.

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Step 1.15.1

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

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

Step 1.15.2

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

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

Step 1.15.3

Cancel the common factor of $2$ .

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Step 1.15.3.1

Cancel the common factor.

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

Step 1.15.3.2

Rewrite the expression.

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

Step 1.15.4

Evaluate the exponent.

$\frac{1}{4}$

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 $(16, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = \frac{1}{4} \cdot (x - (16))$

Step 2.2

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

$y - 8 = \frac{1}{4} \cdot (x - 16)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{4} \cdot (x - 16)$ .

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Step 2.3.1.1

Rewrite.

$y - 8 = 0 + 0 + \frac{1}{4} \cdot (x - 16)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 8 = \frac{1}{4} \cdot (x - 16)$

Step 2.3.1.3

Apply the distributive property.

$y - 8 = \frac{1}{4} x + \frac{1}{4} \cdot - 16$

Step 2.3.1.4

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

$y - 8 = \frac{x}{4} + \frac{1}{4} \cdot - 16$

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 $- 16$ .

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

Step 2.3.1.5.2

Cancel the common factor.

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

Step 2.3.1.5.3

Rewrite the expression.

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

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.

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

Step 2.3.2.2

Add $- 4$ and $8$ .

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

Step 2.3.3

Reorder terms.

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

Step 3