Calculus Examples

$y = \frac{x}{(1 + x^{2})}$ , $(3, 0.3)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = 0.3$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = 1 + x^{2}$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

Multiply $1 + x^{2}$ by $1$ .

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

Multiply $2$ by $- 1$ .

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

Step 1.3

Raise $x$ to the power of $1$ .

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

Step 1.4

Raise $x$ to the power of $1$ .

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

Step 1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.6

Add $1$ and $1$ .

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

Step 1.7

Subtract $2 x^{2}$ from $x^{2}$ .

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

Step 1.8

Reorder terms.

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

Step 1.9

Evaluate the derivative at $x = 3$ .

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

Step 1.10

Simplify.

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Step 1.10.1

Simplify the numerator.

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Step 1.10.1.1

Raise $3$ to the power of $2$ .

$\frac{- 1 \cdot 9 + 1}{{(3^{2} + 1)}^{2}}$

Step 1.10.1.2

Multiply $- 1$ by $9$ .

$\frac{- 9 + 1}{{(3^{2} + 1)}^{2}}$

Step 1.10.1.3

Add $- 9$ and $1$ .

$\frac{- 8}{{(3^{2} + 1)}^{2}}$

Step 1.10.2

Simplify the denominator.

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Step 1.10.2.1

Raise $3$ to the power of $2$ .

$\frac{- 8}{{(9 + 1)}^{2}}$

Step 1.10.2.2

Add $9$ and $1$ .

$\frac{- 8}{10^{2}}$

Step 1.10.2.3

Raise $10$ to the power of $2$ .

$\frac{- 8}{100}$

Step 1.10.3

Reduce the expression by cancelling the common factors.

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Step 1.10.3.1

Cancel the common factor of $- 8$ and $100$ .

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Step 1.10.3.1.1

Factor $4$ out of $- 8$ .

$\frac{4 (- 2)}{100}$

Step 1.10.3.1.2

Cancel the common factors.

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Step 1.10.3.1.2.1

Factor $4$ out of $100$ .

$\frac{4 \cdot - 2}{4 \cdot 25}$

Step 1.10.3.1.2.2

Cancel the common factor.

$\frac{4 \cdot - 2}{4 \cdot 25}$

Step 1.10.3.1.2.3

Rewrite the expression.

$\frac{- 2}{25}$

Step 1.10.3.2

Move the negative in front of the fraction.

$- \frac{2}{25}$

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 $- \frac{2}{25}$ and a given point $(3, 0.3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0.3) = - \frac{2}{25} \cdot (x - (3))$

Step 2.2

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

$y - 0.3 = - \frac{2}{25} \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{2}{25} \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

$y - 0.3 = 0 + 0 - \frac{2}{25} \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 0.3 = - \frac{2}{25} \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y - 0.3 = - \frac{2}{25} x - \frac{2}{25} \cdot - 3$

Step 2.3.1.4

Combine $x$ and $\frac{2}{25}$ .

$y - 0.3 = - \frac{x \cdot 2}{25} - \frac{2}{25} \cdot - 3$

Step 2.3.1.5

Multiply $- \frac{2}{25} \cdot - 3$ .

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Step 2.3.1.5.1

Multiply $- 3$ by $- 1$ .

$y - 0.3 = - \frac{x \cdot 2}{25} + 3 (\frac{2}{25})$

Step 2.3.1.5.2

Combine $3$ and $\frac{2}{25}$ .

$y - 0.3 = - \frac{x \cdot 2}{25} + \frac{3 \cdot 2}{25}$

Step 2.3.1.5.3

Multiply $3$ by $2$ .

$y - 0.3 = - \frac{x \cdot 2}{25} + \frac{6}{25}$

Step 2.3.1.6

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

$y - 0.3 = - \frac{2 x}{25} + \frac{6}{25}$

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 $0.3$ to both sides of the equation.

$y = - \frac{2 x}{25} + \frac{6}{25} + 0.3$

Step 2.3.2.2

To write $0.3$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$y = - \frac{2 x}{25} + \frac{6}{25} + 0.3 \cdot \frac{25}{25}$

Step 2.3.2.3

Combine $0.3$ and $\frac{25}{25}$ .

$y = - \frac{2 x}{25} + \frac{6}{25} + \frac{0.3 \cdot 25}{25}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{2 x}{25} + \frac{6 + 0.3 \cdot 25}{25}$

Step 2.3.2.5

Combine the numerators over the common denominator.

$y = \frac{- 2 x + 6 + 0.3 \cdot 25}{25}$

Step 2.3.2.6

Multiply $0.3$ by $25$ .

$y = \frac{- 2 x + 6 + 7.5}{25}$

Step 2.3.2.7

Add $6$ and $7.5$ .

$y = \frac{- 2 x + 13.5}{25}$

Step 2.3.2.8

Factor $0.5$ out of $- 2 x + 13.5$ .

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Step 2.3.2.8.1

Factor $0.5$ out of $- 2 x$ .

$y = \frac{0.5 (- 4 x) + 13.5}{25}$

Step 2.3.2.8.2

Factor $0.5$ out of $13.5$ .

$y = \frac{0.5 (- 4 x) + 0.5 (27)}{25}$

Step 2.3.2.8.3

Factor $0.5$ out of $0.5 (- 4 x) + 0.5 (27)$ .

$y = \frac{0.5 (- 4 x + 27)}{25}$

Step 2.3.2.9

Factor $25$ out of $25$ .

$y = \frac{0.5 (- 4 x + 27)}{25 (1)}$

Step 2.3.2.10

Separate fractions.

$y = \frac{0.5}{25} \cdot \frac{- 4 x + 27}{1}$

Step 2.3.2.11

Divide $0.5$ by $25$ .

$y = 0.02 \frac{- 4 x + 27}{1}$

Step 2.3.2.12

Cancel the common factor of $0.02$ .

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Step 2.3.2.12.1

Factor $0.02$ out of $1$ .

$y = 0.02 \frac{- 4 x + 27}{0.02 \cdot 50}$

Step 2.3.2.12.2

Cancel the common factor.

$y = 0.02 \frac{- 4 x + 27}{0.02 \cdot 50}$

Step 2.3.2.12.3

Rewrite the expression.

$y = \frac{- 4 x + 27}{50}$

Step 2.3.2.13

Split the fraction $\frac{- 4 x + 27}{50}$ into two fractions.

$y = \frac{- 4 x}{50} + \frac{27}{50}$

Step 2.3.2.14

Simplify each term.

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Step 2.3.2.14.1

Cancel the common factor of $- 4$ and $50$ .

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Step 2.3.2.14.1.1

Factor $2$ out of $- 4 x$ .

$y = \frac{2 (- 2 x)}{50} + \frac{27}{50}$

Step 2.3.2.14.1.2

Cancel the common factors.

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Step 2.3.2.14.1.2.1

Factor $2$ out of $50$ .

$y = \frac{2 (- 2 x)}{2 (25)} + \frac{27}{50}$

Step 2.3.2.14.1.2.2

Cancel the common factor.

$y = \frac{2 (- 2 x)}{2 \cdot 25} + \frac{27}{50}$

Step 2.3.2.14.1.2.3

Rewrite the expression.

$y = \frac{- 2 x}{25} + \frac{27}{50}$

Step 2.3.2.14.2

Move the negative in front of the fraction.

$y = - \frac{2 x}{25} + \frac{27}{50}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{2}{25} x) + \frac{27}{50}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{2}{25} x + \frac{27}{50}$

Step 3