Calculus Examples

$y = \sqrt{1 - 3 x}$ , $(- 1, 2)$

Step 1

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

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

Step 1.2

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 $g (x) = 1 - 3 x$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $1 - 3 x$ .

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

Step 1.2.2

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

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

Step 1.2.3

Replace all occurrences of $u$ with $1 - 3 x$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.6.2

Subtract $2$ from $1$ .

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

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

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

Step 1.7.2

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

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

Step 1.7.3

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

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

Step 1.8

By the Sum Rule, the derivative of $1 - 3 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 3 x]$ .

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

Step 1.9

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.10

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

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

Step 1.11

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

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

Step 1.12

Combine fractions.

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Step 1.12.1

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

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

Step 1.12.2

Move the negative in front of the fraction.

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

Step 1.13

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

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

Step 1.14

Multiply $- 1$ by $1$ .

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

Step 1.15

Evaluate the derivative at $x = - 1$ .

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

Step 1.16

Simplify.

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Step 1.16.1

Simplify the denominator.

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Step 1.16.1.1

Multiply $- 3$ by $- 1$ .

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

Step 1.16.1.2

Add $1$ and $3$ .

$- \frac{3}{2 \cdot 4^{\frac{1}{2}}}$

Step 1.16.1.3

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

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

Step 1.16.1.4

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

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

Step 1.16.1.5

Cancel the common factor of $2$ .

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Step 1.16.1.5.1

Cancel the common factor.

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

Step 1.16.1.5.2

Rewrite the expression.

$- \frac{3}{2 \cdot 2^{1}}$

Step 1.16.1.6

Evaluate the exponent.

$- \frac{3}{2 \cdot 2}$

Step 1.16.2

Multiply $2$ by $2$ .

$- \frac{3}{4}$

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 $- \frac{3}{4}$ and a given point $(- 1, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - \frac{3}{4} \cdot (x - (- 1))$

Step 2.2

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

$y - 2 = - \frac{3}{4} \cdot (x + 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{3}{4} \cdot (x + 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 - \frac{3}{4} \cdot (x + 1)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y - 2 = - \frac{3}{4} x - \frac{3}{4} \cdot 1$

Step 2.3.1.2.2

Combine $x$ and $\frac{3}{4}$ .

$y - 2 = - \frac{x \cdot 3}{4} - \frac{3}{4} \cdot 1$

Step 2.3.1.2.3

Multiply $- 1$ by $1$ .

$y - 2 = - \frac{x \cdot 3}{4} - \frac{3}{4}$

Step 2.3.1.3

Move $3$ to the left of $x$ .

$y - 2 = - \frac{3 x}{4} - \frac{3}{4}$

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 $2$ to both sides of the equation.

$y = - \frac{3 x}{4} - \frac{3}{4} + 2$

Step 2.3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - \frac{3 x}{4} - \frac{3}{4} + 2 \cdot \frac{4}{4}$

Step 2.3.2.3

Combine $2$ and $\frac{4}{4}$ .

$y = - \frac{3 x}{4} - \frac{3}{4} + \frac{2 \cdot 4}{4}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{3 x}{4} + \frac{- 3 + 2 \cdot 4}{4}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $2$ by $4$ .

$y = - \frac{3 x}{4} + \frac{- 3 + 8}{4}$

Step 2.3.2.5.2

Add $- 3$ and $8$ .

$y = - \frac{3 x}{4} + \frac{5}{4}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{3}{4} x) + \frac{5}{4}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{3}{4} x + \frac{5}{4}$

Step 3