Calculus Examples

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

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

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

$\frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [\frac{2}{x}]$

Step 1.2

Evaluate $\frac{d}{d x} [- 3 x^{3}]$ .

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Step 1.2.1

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

$- 3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [\frac{2}{x}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 1.2.3

Multiply $3$ by $- 3$ .

$- 9 x^{2} + \frac{d}{d x} [- \sqrt{x}] + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3

Evaluate $\frac{d}{d x} [- \sqrt{x}]$ .

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Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$- 9 x^{2} + \frac{d}{d x} [- x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$- 9 x^{2} - \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3.3

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

$- 9 x^{2} - (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3.4

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

$- 9 x^{2} - (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3.5

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

$- 9 x^{2} - (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3.6

Combine the numerators over the common denominator.

$- 9 x^{2} - (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3.7

Simplify the numerator.

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Step 1.3.7.1

Multiply $- 1$ by $2$ .

$- 9 x^{2} - (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3.7.2

Subtract $2$ from $1$ .

$- 9 x^{2} - (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3.8

Move the negative in front of the fraction.

$- 9 x^{2} - (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3.9

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

$- 9 x^{2} - \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [\frac{2}{x}]$

Step 1.3.10

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

$- 9 x^{2} - \frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [\frac{2}{x}]$

Step 1.4

Evaluate $\frac{d}{d x} [\frac{2}{x}]$ .

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Step 1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x}]$ .

$- 9 x^{2} - \frac{1}{2 x^{\frac{1}{2}}} + 2 \frac{d}{d x} [\frac{1}{x}]$

Step 1.4.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$- 9 x^{2} - \frac{1}{2 x^{\frac{1}{2}}} + 2 \frac{d}{d x} [x^{- 1}]$

Step 1.4.3

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

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

Step 1.4.4

Multiply $- 1$ by $2$ .

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

Step 1.5

Simplify.

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Step 1.5.1

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

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

Step 1.5.2

Combine terms.

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Step 1.5.2.1

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

$- 9 x^{2} - \frac{1}{2 x^{\frac{1}{2}}} + \frac{- 2}{x^{2}}$

Step 1.5.2.2

Move the negative in front of the fraction.

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

Step 1.5.3

Reorder terms.

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

Step 1.6

Evaluate the derivative at $x = 1$ .

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

Step 1.7

Simplify.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

One to any power is one.

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

Step 1.7.1.2

Multiply $- 9$ by $1$ .

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

Step 1.7.1.3

One to any power is one.

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

Step 1.7.1.4

Divide $2$ by $1$ .

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

Step 1.7.1.5

Multiply $- 1$ by $2$ .

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

Step 1.7.1.6

One to any power is one.

$- 9 - 2 - \frac{1}{2 \cdot 1}$

Step 1.7.1.7

Multiply $2$ by $1$ .

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

Step 1.7.2

Find the common denominator.

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Step 1.7.2.1

Write $- 9$ as a fraction with denominator $1$ .

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

Step 1.7.2.2

Multiply $\frac{- 9}{1}$ by $\frac{2}{2}$ .

$\frac{- 9}{1} \cdot \frac{2}{2} - 2 - \frac{1}{2}$

Step 1.7.2.3

Multiply $\frac{- 9}{1}$ by $\frac{2}{2}$ .

$\frac{- 9 \cdot 2}{2} - 2 - \frac{1}{2}$

Step 1.7.2.4

Write $- 2$ as a fraction with denominator $1$ .

$\frac{- 9 \cdot 2}{2} + \frac{- 2}{1} - \frac{1}{2}$

Step 1.7.2.5

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

$\frac{- 9 \cdot 2}{2} + \frac{- 2}{1} \cdot \frac{2}{2} - \frac{1}{2}$

Step 1.7.2.6

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

$\frac{- 9 \cdot 2}{2} + \frac{- 2 \cdot 2}{2} - \frac{1}{2}$

Step 1.7.3

Combine the numerators over the common denominator.

$\frac{- 9 \cdot 2 - 2 \cdot 2 - 1}{2}$

Step 1.7.4

Simplify each term.

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Step 1.7.4.1

Multiply $- 9$ by $2$ .

$\frac{- 18 - 2 \cdot 2 - 1}{2}$

Step 1.7.4.2

Multiply $- 2$ by $2$ .

$\frac{- 18 - 4 - 1}{2}$

Step 1.7.5

Simplify the expression.

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Step 1.7.5.1

Subtract $4$ from $- 18$ .

$\frac{- 22 - 1}{2}$

Step 1.7.5.2

Subtract $1$ from $- 22$ .

$\frac{- 23}{2}$

Step 1.7.5.3

Move the negative in front of the fraction.

$- \frac{23}{2}$

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 $- \frac{23}{2}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = - \frac{23}{2} \cdot (x - (1))$

Step 2.2

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

$y + 2 = - \frac{23}{2} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{23}{2} \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y + 2 = 0 + 0 - \frac{23}{2} \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 2 = - \frac{23}{2} \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y + 2 = - \frac{23}{2} x - \frac{23}{2} \cdot - 1$

Step 2.3.1.4

Combine $x$ and $\frac{23}{2}$ .

$y + 2 = - \frac{x \cdot 23}{2} - \frac{23}{2} \cdot - 1$

Step 2.3.1.5

Multiply $- \frac{23}{2} \cdot - 1$ .

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Step 2.3.1.5.1

Multiply $- 1$ by $- 1$ .

$y + 2 = - \frac{x \cdot 23}{2} + 1 (\frac{23}{2})$

Step 2.3.1.5.2

Multiply $\frac{23}{2}$ by $1$ .

$y + 2 = - \frac{x \cdot 23}{2} + \frac{23}{2}$

Step 2.3.1.6

Move $23$ to the left of $x$ .

$y + 2 = - \frac{23 x}{2} + \frac{23}{2}$

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

Subtract $2$ from both sides of the equation.

$y = - \frac{23 x}{2} + \frac{23}{2} - 2$

Step 2.3.2.2

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

$y = - \frac{23 x}{2} + \frac{23}{2} - 2 \cdot \frac{2}{2}$

Step 2.3.2.3

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

$y = - \frac{23 x}{2} + \frac{23}{2} + \frac{- 2 \cdot 2}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{23 x}{2} + \frac{23 - 2 \cdot 2}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $- 2$ by $2$ .

$y = - \frac{23 x}{2} + \frac{23 - 4}{2}$

Step 2.3.2.5.2

Subtract $4$ from $23$ .

$y = - \frac{23 x}{2} + \frac{19}{2}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{23}{2} x) + \frac{19}{2}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{23}{2} x + \frac{19}{2}$

Step 3