Calculus Examples

$f (x) = \frac{1}{8} x^{3}$ at $x = 3$

Step 1

Find the corresponding $y$ -value to $x = 3$ .

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Step 1.1

Substitute $3$ in for $x$ .

$y = \frac{1}{8} \cdot {(3)}^{3}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \frac{1}{8} \cdot 3^{3}$

Step 1.2.2

Remove parentheses.

$y = \frac{1}{8} \cdot {(3)}^{3}$

Step 1.2.3

Simplify $\frac{1}{8} \cdot {(3)}^{3}$ .

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Step 1.2.3.1

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

$y = \frac{1}{8} \cdot 27$

Step 1.2.3.2

Combine $\frac{1}{8}$ and $27$ .

$y = \frac{27}{8}$

Step 2

Find the first derivative and evaluate at $x = 3$ and $y = \frac{27}{8}$ to find the slope of the tangent line.

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Step 2.1

Since $\frac{1}{8}$ is constant with respect to $x$ , the derivative of $\frac{1}{8} x^{3}$ with respect to $x$ is $\frac{1}{8} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{8} \frac{d}{d x} [x^{3}]$

Step 2.2

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

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

Step 2.3

Combine fractions.

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Step 2.3.1

Combine $3$ and $\frac{1}{8}$ .

$\frac{3}{8} x^{2}$

Step 2.3.2

Combine $\frac{3}{8}$ and $x^{2}$ .

$\frac{3 x^{2}}{8}$

Step 2.4

Evaluate the derivative at $x = 3$ .

$\frac{3 {(3)}^{2}}{8}$

Step 2.5

Simplify.

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Step 2.5.1

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 2.5.1.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 2.5.1.1.1

Raise $3$ to the power of $1$ .

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

Step 2.5.1.1.2

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

$\frac{3^{1 + 2}}{8}$

Step 2.5.1.2

Add $1$ and $2$ .

$\frac{3^{3}}{8}$

Step 2.5.2

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

$\frac{27}{8}$

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 $\frac{27}{8}$ and a given point $(3, \frac{27}{8})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{27}{8}) = \frac{27}{8} \cdot (x - (3))$

Step 3.2

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

$y - \frac{27}{8} = \frac{27}{8} \cdot (x - 3)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{27}{8} \cdot (x - 3)$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{27}{8} = 0 + 0 + \frac{27}{8} \cdot (x - 3)$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{27}{8} = \frac{27}{8} \cdot (x - 3)$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{27}{8} = \frac{27}{8} x + \frac{27}{8} \cdot - 3$

Step 3.3.1.4

Combine $\frac{27}{8}$ and $x$ .

$y - \frac{27}{8} = \frac{27 x}{8} + \frac{27}{8} \cdot - 3$

Step 3.3.1.5

Multiply $\frac{27}{8} \cdot - 3$ .

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Step 3.3.1.5.1

Combine $\frac{27}{8}$ and $- 3$ .

$y - \frac{27}{8} = \frac{27 x}{8} + \frac{27 \cdot - 3}{8}$

Step 3.3.1.5.2

Multiply $27$ by $- 3$ .

$y - \frac{27}{8} = \frac{27 x}{8} + \frac{- 81}{8}$

Step 3.3.1.6

Move the negative in front of the fraction.

$y - \frac{27}{8} = \frac{27 x}{8} - \frac{81}{8}$

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 $\frac{27}{8}$ to both sides of the equation.

$y = \frac{27 x}{8} - \frac{81}{8} + \frac{27}{8}$

Step 3.3.2.2

Combine the numerators over the common denominator.

$y = \frac{27 x - 81 + 27}{8}$

Step 3.3.2.3

Add $- 81$ and $27$ .

$y = \frac{27 x - 54}{8}$

Step 3.3.2.4

Factor $27$ out of $27 x - 54$ .

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Step 3.3.2.4.1

Factor $27$ out of $27 x$ .

$y = \frac{27 (x) - 54}{8}$

Step 3.3.2.4.2

Factor $27$ out of $- 54$ .

$y = \frac{27 x + 27 \cdot - 2}{8}$

Step 3.3.2.4.3

Factor $27$ out of $27 x + 27 \cdot - 2$ .

$y = \frac{27 (x - 2)}{8}$

Step 3.3.3

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

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Step 3.3.3.1

Apply the distributive property.

$y = \frac{27 x + 27 \cdot - 2}{8}$

Step 3.3.3.2

Multiply $27$ by $- 2$ .

$y = \frac{27 x - 54}{8}$

Step 3.3.3.3

Split the fraction $\frac{27 x - 54}{8}$ into two fractions.

$y = \frac{27 x}{8} + \frac{- 54}{8}$

Step 3.3.3.4

Cancel the common factor of $- 54$ and $8$ .

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Step 3.3.3.4.1

Factor $2$ out of $- 54$ .

$y = \frac{27 x}{8} + \frac{2 (- 27)}{8}$

Step 3.3.3.4.2

Cancel the common factors.

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Step 3.3.3.4.2.1

Factor $2$ out of $8$ .

$y = \frac{27 x}{8} + \frac{2 \cdot - 27}{2 \cdot 4}$

Step 3.3.3.4.2.2

Cancel the common factor.

$y = \frac{27 x}{8} + \frac{2 \cdot - 27}{2 \cdot 4}$

Step 3.3.3.4.2.3

Rewrite the expression.

$y = \frac{27 x}{8} + \frac{- 27}{4}$

Step 3.3.3.5

Move the negative in front of the fraction.

$y = \frac{27 x}{8} - \frac{27}{4}$

Step 3.3.3.6

Reorder terms.

$y = \frac{27}{8} x - \frac{27}{4}$

Step 4