Calculus Examples

$f (x) = \sqrt{4 x + 36}$ ; $x = 0$

Step 1

Find the corresponding $y$ -value to $x = 0$ .

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Step 1.1

Substitute $0$ in for $x$ .

$y = \sqrt{4 (0) + 36}$

Step 1.2

Simplify $\sqrt{4 (0) + 36}$ .

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Step 1.2.1

Multiply $4$ by $0$ .

$y = \sqrt{0 + 36}$

Step 1.2.2

Add $0$ and $36$ .

$y = \sqrt{36}$

Step 1.2.3

Rewrite $36$ as $6^{2}$ .

$y = \sqrt{6^{2}}$

Step 1.2.4

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

$y = 6$

Step 2

Find the first derivative and evaluate at $x = 0$ and $y = 6$ to find the slope of the tangent line.

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Step 2.1

Rewrite $\frac{d}{d x} [\sqrt{4 x + 36}]$ as $\frac{d}{d x} [2 \sqrt{x + 9}]$ .

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Step 2.1.1

Rewrite $4 x + 36$ as $2^{2} (x + 9)$ .

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Step 2.1.1.1

Factor $4$ out of $4 x$ .

$\frac{d}{d x} [\sqrt{4 (x) + 36}]$

Step 2.1.1.2

Factor $4$ out of $36$ .

$\frac{d}{d x} [\sqrt{4 (x) + 4 (9)}]$

Step 2.1.1.3

Factor $4$ out of $4 (x) + 4 (9)$ .

$\frac{d}{d x} [\sqrt{4 (x + 9)}]$

Step 2.1.1.4

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

$\frac{d}{d x} [\sqrt{2^{2} (x + 9)}]$

Step 2.1.2

Pull terms out from under the radical.

$\frac{d}{d x} [2 \sqrt{x + 9}]$

Step 2.2

Differentiate using the Constant Multiple Rule.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 9}$ as ${(x + 9)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [2 {(x + 9)}^{\frac{1}{2}}]$

Step 2.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 {(x + 9)}^{\frac{1}{2}}$ with respect to $x$ is $2 \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]$ .

$2 \frac{d}{d x} [{(x + 9)}^{\frac{1}{2}}]$

Step 2.3

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 $g (x) = x + 9$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $x + 9$ .

$2 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x + 9])$

Step 2.3.2

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

$2 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x + 9])$

Step 2.3.3

Replace all occurrences of $u$ with $x + 9$ .

$2 (\frac{1}{2} {(x + 9)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 9])$

Step 2.4

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

$2 (\frac{1}{2} {(x + 9)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x + 9])$

Step 2.5

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

$2 (\frac{1}{2} {(x + 9)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x + 9])$

Step 2.6

Combine the numerators over the common denominator.

$2 (\frac{1}{2} {(x + 9)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x + 9])$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

$2 (\frac{1}{2} {(x + 9)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x + 9])$

Step 2.7.2

Subtract $2$ from $1$ .

$2 (\frac{1}{2} {(x + 9)}^{\frac{- 1}{2}} \frac{d}{d x} [x + 9])$

Step 2.8

Simplify terms.

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Step 2.8.1

Move the negative in front of the fraction.

$2 (\frac{1}{2} {(x + 9)}^{- \frac{1}{2}} \frac{d}{d x} [x + 9])$

Step 2.8.2

Combine $\frac{1}{2}$ and ${(x + 9)}^{- \frac{1}{2}}$ .

$2 (\frac{{(x + 9)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x + 9])$

Step 2.8.3

Move ${(x + 9)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 (\frac{1}{2 {(x + 9)}^{\frac{1}{2}}} \frac{d}{d x} [x + 9])$

Step 2.8.4

Combine $\frac{1}{2 {(x + 9)}^{\frac{1}{2}}}$ and $2$ .

$\frac{2}{2 {(x + 9)}^{\frac{1}{2}}} \frac{d}{d x} [x + 9]$

Step 2.8.5

Cancel the common factor.

$\frac{2}{2 {(x + 9)}^{\frac{1}{2}}} \frac{d}{d x} [x + 9]$

Step 2.8.6

Rewrite the expression.

$\frac{1}{{(x + 9)}^{\frac{1}{2}}} \frac{d}{d x} [x + 9]$

Step 2.9

By the Sum Rule, the derivative of $x + 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [9]$ .

$\frac{1}{{(x + 9)}^{\frac{1}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [9])$

Step 2.10

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

$\frac{1}{{(x + 9)}^{\frac{1}{2}}} (1 + \frac{d}{d x} [9])$

Step 2.11

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{1}{{(x + 9)}^{\frac{1}{2}}} (1 + 0)$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $1$ and $0$ .

$\frac{1}{{(x + 9)}^{\frac{1}{2}}} \cdot 1$

Step 2.12.2

Multiply $\frac{1}{{(x + 9)}^{\frac{1}{2}}}$ by $1$ .

$\frac{1}{{(x + 9)}^{\frac{1}{2}}}$

Step 2.13

Evaluate the derivative at $x = 0$ .

$\frac{1}{{((0) + 9)}^{\frac{1}{2}}}$

Step 2.14

Simplify the denominator.

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Step 2.14.1

Add $0$ and $9$ .

$\frac{1}{9^{\frac{1}{2}}}$

Step 2.14.2

Rewrite $9$ as $3^{2}$ .

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

Step 2.14.3

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

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

Step 2.14.4

Cancel the common factor of $2$ .

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Step 2.14.4.1

Cancel the common factor.

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

Step 2.14.4.2

Rewrite the expression.

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

Step 2.14.5

Evaluate the exponent.

$\frac{1}{3}$

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 $\frac{1}{3}$ and a given point $(0, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = \frac{1}{3} \cdot (x - (0))$

Step 3.2

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

$y - 6 = \frac{1}{3} \cdot (x + 0)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{1}{3} \cdot (x + 0)$ .

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Step 3.3.1.1

Add $x$ and $0$ .

$y - 6 = \frac{1}{3} \cdot x$

Step 3.3.1.2

Combine $\frac{1}{3}$ and $x$ .

$y - 6 = \frac{x}{3}$

Step 3.3.2

Add $6$ to both sides of the equation.

$y = \frac{x}{3} + 6$

Step 3.3.3

Reorder terms.

$y = \frac{1}{3} x + 6$

Step 4