Calculus Examples

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

Step 1

Find the corresponding $y$ -value to $x = 1$ .

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Step 1.1

Substitute $1$ in for $x$ .

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

Step 1.2

Simplify $\sqrt{2 - (1)}$ .

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Step 1.2.1

Multiply $- 1$ by $1$ .

$y = \sqrt{2 - 1}$

Step 1.2.2

Subtract $1$ from $2$ .

$y = \sqrt{1}$

Step 1.2.3

Any root of $1$ is $1$ .

$y = 1$

Step 2

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

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

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 $g (x) = 2 - x$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $1$ .

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

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

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

Step 2.7.2

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

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

Step 2.7.3

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

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

Step 2.8

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

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

Step 2.9

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

Combine fractions.

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Step 2.13.1

Multiply $- 1$ by $1$ .

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

Step 2.13.2

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

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

Step 2.13.3

Move the negative in front of the fraction.

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

Step 2.14

Evaluate the derivative at $x = 1$ .

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

Step 2.15

Simplify.

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Step 2.15.1

Simplify the denominator.

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Step 2.15.1.1

Multiply $- 1$ by $1$ .

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

Step 2.15.1.2

Subtract $1$ from $2$ .

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

Step 2.15.1.3

One to any power is one.

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

Step 2.15.2

Multiply $2$ by $1$ .

$- \frac{1}{2}$

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

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

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite.

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

Step 3.3.1.2

Simplify by adding zeros.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.1.4

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

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

Step 3.3.1.5

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

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Step 3.3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.5.2

Multiply $\frac{1}{2}$ by $1$ .

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

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 $1$ to both sides of the equation.

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

Step 3.3.2.2

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

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

Step 3.3.2.3

Combine the numerators over the common denominator.

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

Step 3.3.2.4

Add $1$ and $2$ .

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

Step 3.3.3

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

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Step 3.3.3.1

Reorder terms.

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

Step 3.3.3.2

Remove parentheses.

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

Step 4