Calculus Examples

$f (x) = \sqrt{x^{2} + 3}$ , $x = - 2$

Step 1

Find the corresponding $y$ -value to $x = - 2$ .

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Step 1.1

Substitute $- 2$ in for $x$ .

$y = \sqrt{{(- 2)}^{2} + 3}$

Step 1.2

Simplify $\sqrt{{(- 2)}^{2} + 3}$ .

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Step 1.2.1

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

$y = \sqrt{4 + 3}$

Step 1.2.2

Add $4$ and $3$ .

$y = \sqrt{7}$

Step 2

Find the first derivative and evaluate at $x = - 2$ and $y = \sqrt{7}$ 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 $\sqrt{x^{2} + 3}$ as ${(x^{2} + 3)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{2} + 3)}^{\frac{1}{2}}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{2} + 3$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 3$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} + 3]$

Step 2.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 3]$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} + 3$ .

$\frac{1}{2} {(x^{2} + 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 3]$

Step 2.3

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

$\frac{1}{2} {(x^{2} + 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} + 3]$

Step 2.4

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

$\frac{1}{2} {(x^{2} + 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 3]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{2} + 3)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 3]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{2} + 3)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} + 3]$

Step 2.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{2} + 3)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} + 3]$

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{2} + 3)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} + 3]$

Step 2.7.2

Combine $\frac{1}{2}$ and ${(x^{2} + 3)}^{- \frac{1}{2}}$ .

$\frac{{(x^{2} + 3)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} + 3]$

Step 2.7.3

Move ${(x^{2} + 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{2} + 3)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} + 3]$

Step 2.8

By the Sum Rule, the derivative of $x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$ .

$\frac{1}{2 {(x^{2} + 3)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3])$

Step 2.9

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

$\frac{1}{2 {(x^{2} + 3)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [3])$

Step 2.10

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x^{2} + 3)}^{\frac{1}{2}}} (2 x + 0)$

Step 2.11

Simplify terms.

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Step 2.11.1

Add $2 x$ and $0$ .

$\frac{1}{2 {(x^{2} + 3)}^{\frac{1}{2}}} (2 x)$

Step 2.11.2

Combine $2$ and $\frac{1}{2 {(x^{2} + 3)}^{\frac{1}{2}}}$ .

$\frac{2}{2 {(x^{2} + 3)}^{\frac{1}{2}}} x$

Step 2.11.3

Combine $\frac{2}{2 {(x^{2} + 3)}^{\frac{1}{2}}}$ and $x$ .

$\frac{2 x}{2 {(x^{2} + 3)}^{\frac{1}{2}}}$

Step 2.11.4

Cancel the common factor.

$\frac{2 x}{2 {(x^{2} + 3)}^{\frac{1}{2}}}$

Step 2.11.5

Rewrite the expression.

$\frac{x}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 2.12

Evaluate the derivative at $x = - 2$ .

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

Step 2.13

Simplify.

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Step 2.13.1

Simplify the denominator.

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Step 2.13.1.1

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

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

Step 2.13.1.2

Add $4$ and $3$ .

$\frac{- 2}{7^{\frac{1}{2}}}$

Step 2.13.2

Move the negative in front of the fraction.

$- \frac{2}{7^{\frac{1}{2}}}$

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 $- \frac{2}{7^{\frac{1}{2}}}$ and a given point $(- 2, \sqrt{7})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\sqrt{7}) = - \frac{2}{7^{\frac{1}{2}}} \cdot (x - (- 2))$

Step 3.2

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

$y - \sqrt{7} = - \frac{2}{7^{\frac{1}{2}}} \cdot (x + 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- \frac{2}{7^{\frac{1}{2}}} \cdot (x + 2)$ .

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Step 3.3.1.1

Rewrite.

$y - \sqrt{7} = 0 + 0 - \frac{2}{7^{\frac{1}{2}}} \cdot (x + 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y - \sqrt{7} = - \frac{2}{7^{\frac{1}{2}}} \cdot (x + 2)$

Step 3.3.1.3

Apply the distributive property.

$y - \sqrt{7} = - \frac{2}{7^{\frac{1}{2}}} x - \frac{2}{7^{\frac{1}{2}}} \cdot 2$

Step 3.3.1.4

Combine $x$ and $\frac{2}{7^{\frac{1}{2}}}$ .

$y - \sqrt{7} = - \frac{x \cdot 2}{7^{\frac{1}{2}}} - \frac{2}{7^{\frac{1}{2}}} \cdot 2$

Step 3.3.1.5

Multiply $- \frac{2}{7^{\frac{1}{2}}} \cdot 2$ .

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Step 3.3.1.5.1

Multiply $2$ by $- 1$ .

$y - \sqrt{7} = - \frac{x \cdot 2}{7^{\frac{1}{2}}} - 2 \frac{2}{7^{\frac{1}{2}}}$

Step 3.3.1.5.2

Combine $- 2$ and $\frac{2}{7^{\frac{1}{2}}}$ .

$y - \sqrt{7} = - \frac{x \cdot 2}{7^{\frac{1}{2}}} + \frac{- 2 \cdot 2}{7^{\frac{1}{2}}}$

Step 3.3.1.5.3

Multiply $- 2$ by $2$ .

$y - \sqrt{7} = - \frac{x \cdot 2}{7^{\frac{1}{2}}} + \frac{- 4}{7^{\frac{1}{2}}}$

Step 3.3.1.6

Simplify each term.

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Step 3.3.1.6.1

Move $2$ to the left of $x$ .

$y - \sqrt{7} = - \frac{2 \cdot x}{7^{\frac{1}{2}}} + \frac{- 4}{7^{\frac{1}{2}}}$

Step 3.3.1.6.2

Move the negative in front of the fraction.

$y - \sqrt{7} = - \frac{2 x}{7^{\frac{1}{2}}} - \frac{4}{7^{\frac{1}{2}}}$

Step 3.3.2

Add $\sqrt{7}$ to both sides of the equation.

$y = - \frac{2 x}{7^{\frac{1}{2}}} - \frac{4}{7^{\frac{1}{2}}} + \sqrt{7}$

Step 3.3.3

Write in $y = m x + b$ form.

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Step 3.3.3.1

To write $\sqrt{7}$ as a fraction with a common denominator, multiply by $\frac{7^{\frac{1}{2}}}{7^{\frac{1}{2}}}$ .

$y = - \frac{2 x}{7^{\frac{1}{2}}} - \frac{4}{7^{\frac{1}{2}}} + \sqrt{7} \cdot \frac{7^{\frac{1}{2}}}{7^{\frac{1}{2}}}$

Step 3.3.3.2

Combine $\sqrt{7}$ and $\frac{7^{\frac{1}{2}}}{7^{\frac{1}{2}}}$ .

$y = - \frac{2 x}{7^{\frac{1}{2}}} - \frac{4}{7^{\frac{1}{2}}} + \frac{\sqrt{7} \cdot 7^{\frac{1}{2}}}{7^{\frac{1}{2}}}$

Step 3.3.3.3

Combine the numerators over the common denominator.

$y = - \frac{2 x}{7^{\frac{1}{2}}} + \frac{- 4 + \sqrt{7} \cdot 7^{\frac{1}{2}}}{7^{\frac{1}{2}}}$

Step 3.3.3.4

Simplify the numerator.

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Step 3.3.3.4.1

Multiply $\sqrt{7} \cdot 7^{\frac{1}{2}}$ .

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Step 3.3.3.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = - \frac{2 x}{7^{\frac{1}{2}}} + \frac{- 4 + 7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}}{7^{\frac{1}{2}}}$

Step 3.3.3.4.1.2

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

$y = - \frac{2 x}{7^{\frac{1}{2}}} + \frac{- 4 + 7^{\frac{1}{2} + \frac{1}{2}}}{7^{\frac{1}{2}}}$

Step 3.3.3.4.1.3

Combine the numerators over the common denominator.

$y = - \frac{2 x}{7^{\frac{1}{2}}} + \frac{- 4 + 7^{\frac{1 + 1}{2}}}{7^{\frac{1}{2}}}$

Step 3.3.3.4.1.4

Add $1$ and $1$ .

$y = - \frac{2 x}{7^{\frac{1}{2}}} + \frac{- 4 + 7^{\frac{2}{2}}}{7^{\frac{1}{2}}}$

Step 3.3.3.4.1.5

Cancel the common factor of $2$ .

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Step 3.3.3.4.1.5.1

Cancel the common factor.

$y = - \frac{2 x}{7^{\frac{1}{2}}} + \frac{- 4 + 7^{\frac{2}{2}}}{7^{\frac{1}{2}}}$

Step 3.3.3.4.1.5.2

Rewrite the expression.

$y = - \frac{2 x}{7^{\frac{1}{2}}} + \frac{- 4 + 7^{1}}{7^{\frac{1}{2}}}$

Step 3.3.3.4.2

Evaluate the exponent.

$y = - \frac{2 x}{7^{\frac{1}{2}}} + \frac{- 4 + 7}{7^{\frac{1}{2}}}$

Step 3.3.3.4.3

Add $- 4$ and $7$ .

$y = - \frac{2 x}{7^{\frac{1}{2}}} + \frac{3}{7^{\frac{1}{2}}}$

Step 3.3.3.5

Reorder terms.

$y = - (\frac{2}{7^{\frac{1}{2}}} x) + \frac{3}{7^{\frac{1}{2}}}$

Step 3.3.3.6

Remove parentheses.

$y = - \frac{2}{7^{\frac{1}{2}}} x + \frac{3}{7^{\frac{1}{2}}}$

Step 4