Calculus Examples

$f (x) = \frac{2}{3} x^{2} - x + 1$ at $(2, \frac{5}{3})$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = \frac{5}{3}$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of $\frac{2}{3} x^{2} - x + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{2}{3} x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [1]$ .

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

Step 1.2

Evaluate $\frac{d}{d x} [\frac{2}{3} x^{2}]$ .

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Step 1.2.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2}{3} x^{2}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [x^{2}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $2$ by $2$ .

$\frac{4}{3} x + \frac{d}{d x} [- x] + \frac{d}{d x} [1]$

Step 1.2.5

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

$\frac{4 x}{3} + \frac{d}{d x} [- x] + \frac{d}{d x} [1]$

Step 1.3

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

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Step 1.3.1

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

$\frac{4 x}{3} - \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 1.3.2

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

$\frac{4 x}{3} - 1 \cdot 1 + \frac{d}{d x} [1]$

Step 1.3.3

Multiply $- 1$ by $1$ .

$\frac{4 x}{3} - 1 + \frac{d}{d x} [1]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

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

$\frac{4 x}{3} - 1 + 0$

Step 1.4.2

Add $\frac{4 x}{3} - 1$ and $0$ .

$\frac{4 x}{3} - 1$

Step 1.5

Evaluate the derivative at $x = 2$ .

$\frac{4 (2)}{3} - 1$

Step 1.6

Simplify.

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Step 1.6.1

Multiply $4$ by $2$ .

$\frac{8}{3} - 1$

Step 1.6.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} - 1 \cdot \frac{3}{3}$

Step 1.6.3

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{8}{3} + \frac{- 1 \cdot 3}{3}$

Step 1.6.4

Combine the numerators over the common denominator.

$\frac{8 - 1 \cdot 3}{3}$

Step 1.6.5

Simplify the numerator.

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Step 1.6.5.1

Multiply $- 1$ by $3$ .

$\frac{8 - 3}{3}$

Step 1.6.5.2

Subtract $3$ from $8$ .

$\frac{5}{3}$

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 $\frac{5}{3}$ and a given point $(2, \frac{5}{3})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{5}{3}) = \frac{5}{3} \cdot (x - (2))$

Step 2.2

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

$y - \frac{5}{3} = \frac{5}{3} \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{5}{3} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{5}{3} = 0 + 0 + \frac{5}{3} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{5}{3} = \frac{5}{3} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{5}{3} = \frac{5}{3} x + \frac{5}{3} \cdot - 2$

Step 2.3.1.4

Combine $\frac{5}{3}$ and $x$ .

$y - \frac{5}{3} = \frac{5 x}{3} + \frac{5}{3} \cdot - 2$

Step 2.3.1.5

Multiply $\frac{5}{3} \cdot - 2$ .

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Step 2.3.1.5.1

Combine $\frac{5}{3}$ and $- 2$ .

$y - \frac{5}{3} = \frac{5 x}{3} + \frac{5 \cdot - 2}{3}$

Step 2.3.1.5.2

Multiply $5$ by $- 2$ .

$y - \frac{5}{3} = \frac{5 x}{3} + \frac{- 10}{3}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - \frac{5}{3} = \frac{5 x}{3} - \frac{10}{3}$

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 $\frac{5}{3}$ to both sides of the equation.

$y = \frac{5 x}{3} - \frac{10}{3} + \frac{5}{3}$

Step 2.3.2.2

Combine the numerators over the common denominator.

$y = \frac{5 x - 10 + 5}{3}$

Step 2.3.2.3

Add $- 10$ and $5$ .

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

Step 2.3.2.4

Factor $5$ out of $5 x - 5$ .

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Step 2.3.2.4.1

Factor $5$ out of $5 x$ .

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

Step 2.3.2.4.2

Factor $5$ out of $- 5$ .

$y = \frac{5 (x) + 5 (- 1)}{3}$

Step 2.3.2.4.3

Factor $5$ out of $5 (x) + 5 (- 1)$ .

$y = \frac{5 (x - 1)}{3}$

Step 2.3.3

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

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Step 2.3.3.1

Apply the distributive property.

$y = \frac{5 x + 5 \cdot - 1}{3}$

Step 2.3.3.2

Multiply $5$ by $- 1$ .

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

Step 2.3.3.3

Split the fraction $\frac{5 x - 5}{3}$ into two fractions.

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

Step 2.3.3.4

Move the negative in front of the fraction.

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

Step 2.3.3.5

Reorder terms.

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

Step 3