Calculus Examples

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

Step 1

Write $\frac{1}{1 + x^{2}}$ as an equation.

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

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = \frac{1}{2}$ to find the slope of the tangent line.

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Step 2.1

Rewrite $\frac{1}{1 + x^{2}}$ as ${(1 + x^{2})}^{- 1}$ .

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

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 $f (x) = x^{- 1}$ and $g (x) = 1 + x^{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $1 + x^{2}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $1 + x^{2}$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $1 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$ .

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

Step 2.3.2

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

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

Step 2.3.3

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

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

Step 2.3.4

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

$- {(1 + x^{2})}^{- 2} (2 x)$

Step 2.3.5

Multiply $2$ by $- 1$ .

$- 2 {(1 + x^{2})}^{- 2} x$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.4.2

Combine terms.

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Step 2.4.2.1

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

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

Step 2.4.2.2

Move the negative in front of the fraction.

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

Step 2.4.2.3

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

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

Step 2.4.2.4

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

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

Step 2.5

Evaluate the derivative at $x = 1$ .

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

Step 2.6

Simplify.

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Step 2.6.1

Multiply $2$ by $1$ .

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

Step 2.6.2

Simplify the denominator.

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Step 2.6.2.1

One to any power is one.

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

Step 2.6.2.2

Add $1$ and $1$ .

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

Step 2.6.2.3

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

$- \frac{2}{4}$

Step 2.6.3

Cancel the common factor of $2$ and $4$ .

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Step 2.6.3.1

Factor $2$ out of $2$ .

$- \frac{2 (1)}{4}$

Step 2.6.3.2

Cancel the common factors.

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Step 2.6.3.2.1

Factor $2$ out of $4$ .

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

Step 2.6.3.2.2

Cancel the common factor.

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

Step 2.6.3.2.3

Rewrite the expression.

$- \frac{1}{2}$

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

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

Step 3.2

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

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

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.

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

Step 3.3.1.2

Simplify by adding zeros.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.1.4

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

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

Step 3.3.1.5

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

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Step 3.3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.5.2

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

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

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 $\frac{1}{2}$ to both sides of the equation.

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

Step 3.3.2.2

Combine the numerators over the common denominator.

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

Step 3.3.2.3

Add $1$ and $1$ .

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

Step 3.3.2.4

Split the fraction $\frac{- x + 2}{2}$ into two fractions.

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

Step 3.3.2.5

Simplify each term.

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Step 3.3.2.5.1

Move the negative in front of the fraction.

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

Step 3.3.2.5.2

Divide $2$ by $2$ .

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

Step 3.3.3

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

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Step 3.3.3.1

Reorder terms.

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

Step 3.3.3.2

Remove parentheses.

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

Step 4