Calculus Examples

$y = 8 \sqrt{x}$ , $(4, 16)$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = 16$ 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 $x^{\frac{1}{2}}$ .

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

Step 1.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{\frac{1}{2}}$ with respect to $x$ is $8 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Move the negative in front of the fraction.

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

Factor $2$ out of $8$ .

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

Step 1.13

Cancel the common factors.

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Step 1.13.1

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

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

Step 1.13.2

Cancel the common factor.

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

Step 1.13.3

Rewrite the expression.

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

Step 1.14

Evaluate the derivative at $x = 4$ .

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

Step 1.15

Simplify.

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Step 1.15.1

Move ${(4)}^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 1.15.2

Multiply $4$ by $4^{- \frac{1}{2}}$ by adding the exponents.

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Step 1.15.2.1

Multiply $4$ by $4^{- \frac{1}{2}}$ .

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Step 1.15.2.1.1

Raise $4$ to the power of $1$ .

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

Step 1.15.2.1.2

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

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

Step 1.15.2.2

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

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

Step 1.15.2.3

Combine the numerators over the common denominator.

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

Step 1.15.2.4

Subtract $1$ from $2$ .

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

Step 1.15.3

Simplify the expression.

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Step 1.15.3.1

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

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

Step 1.15.3.2

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

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

Step 1.15.4

Cancel the common factor of $2$ .

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Step 1.15.4.1

Cancel the common factor.

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

Step 1.15.4.2

Rewrite the expression.

$2^{1}$

Step 1.15.5

Evaluate the exponent.

$2$

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 $2$ and a given point $(4, 16)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (16) = 2 \cdot (x - (4))$

Step 2.2

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

$y - 16 = 2 \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $2 \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - 16 = 0 + 0 + 2 \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 16 = 2 \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y - 16 = 2 x + 2 \cdot - 4$

Step 2.3.1.4

Multiply $2$ by $- 4$ .

$y - 16 = 2 x - 8$

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 $16$ to both sides of the equation.

$y = 2 x - 8 + 16$

Step 2.3.2.2

Add $- 8$ and $16$ .

$y = 2 x + 8$

Step 3