Calculus Examples

$f (x) = \sqrt{2 x - 3}$ , $x = 6$

Step 1

Find the corresponding $y$ -value to $x = 6$ .

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Step 1.1

Substitute $6$ in for $x$ .

$y = \sqrt{2 (6) - 3}$

Step 1.2

Simplify $\sqrt{2 (6) - 3}$ .

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Step 1.2.1

Multiply $2$ by $6$ .

$y = \sqrt{12 - 3}$

Step 1.2.2

Subtract $3$ from $12$ .

$y = \sqrt{9}$

Step 1.2.3

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

$y = \sqrt{3^{2}}$

Step 1.2.4

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

$y = 3$

Step 2

Find the first derivative and evaluate at $x = 6$ and $y = 3$ 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 $\sqrt{2 x - 3}$ as ${(2 x - 3)}^{\frac{1}{2}}$ .

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

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 $g (x) = 2 x - 3$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $2 x - 3$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $2 x - 3$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $1$ .

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

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

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

Step 2.7.2

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

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

Step 2.7.3

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Multiply $2$ by $1$ .

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

Step 2.12

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

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

Step 2.13

Simplify terms.

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Step 2.13.1

Add $2$ and $0$ .

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

Step 2.13.2

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

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

Step 2.13.3

Cancel the common factor.

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

Step 2.13.4

Rewrite the expression.

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

Step 2.14

Evaluate the derivative at $x = 6$ .

$\frac{1}{{(2 (6) - 3)}^{\frac{1}{2}}}$

Step 2.15

Simplify the denominator.

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Step 2.15.1

Multiply $2$ by $6$ .

$\frac{1}{{(12 - 3)}^{\frac{1}{2}}}$

Step 2.15.2

Subtract $3$ from $12$ .

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

Step 2.15.3

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

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

Step 2.15.4

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

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

Step 2.15.5

Cancel the common factor of $2$ .

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Step 2.15.5.1

Cancel the common factor.

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

Step 2.15.5.2

Rewrite the expression.

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

Step 2.15.6

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 $(6, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{1}{3} \cdot (x - 6)$ .

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Step 3.3.1.1

Rewrite.

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

Step 3.3.1.2

Simplify by adding zeros.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.1.4

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

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

Step 3.3.1.5

Cancel the common factor of $3$ .

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Step 3.3.1.5.1

Factor $3$ out of $- 6$ .

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

Step 3.3.1.5.2

Cancel the common factor.

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

Step 3.3.1.5.3

Rewrite the expression.

$y - 3 = \frac{x}{3} - 2$

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 $3$ to both sides of the equation.

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

Step 3.3.2.2

Add $- 2$ and $3$ .

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

Step 3.3.3

Reorder terms.

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

Step 4