Calculus Examples

$f (x) = \frac{1}{6} x^{4}$ at $x = - 1$

Step 1

Find the corresponding $y$ -value to $x = - 1$ .

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Step 1.1

Substitute $- 1$ in for $x$ .

$y = \frac{1}{6} \cdot {(- 1)}^{4}$

Step 1.2

Simplify $\frac{1}{6} \cdot {(- 1)}^{4}$ .

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Step 1.2.1

Raise $- 1$ to the power of $4$ .

$y = \frac{1}{6} \cdot 1$

Step 1.2.2

Multiply $\frac{1}{6}$ by $1$ .

$y = \frac{1}{6}$

Step 2

Find the first derivative and evaluate at $x = - 1$ and $y = \frac{1}{6}$ to find the slope of the tangent line.

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Step 2.1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{1}{6} x^{4}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{6} \frac{d}{d x} [x^{4}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{6} (4 x^{3})$

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $4$ and $\frac{1}{6}$ .

$\frac{4}{6} x^{3}$

Step 2.3.2

Combine $\frac{4}{6}$ and $x^{3}$ .

$\frac{4 x^{3}}{6}$

Step 2.3.3

Cancel the common factor of $4$ and $6$ .

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Step 2.3.3.1

Factor $2$ out of $4 x^{3}$ .

$\frac{2 (2 x^{3})}{6}$

Step 2.3.3.2

Cancel the common factors.

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Step 2.3.3.2.1

Factor $2$ out of $6$ .

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

Step 2.3.3.2.2

Cancel the common factor.

$\frac{2 (2 x^{3})}{2 \cdot 3}$

Step 2.3.3.2.3

Rewrite the expression.

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

Step 2.4

Evaluate the derivative at $x = - 1$ .

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

Step 2.5

Simplify.

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Step 2.5.1

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

$\frac{2 \cdot - 1}{3}$

Step 2.5.2

Multiply $2$ by $- 1$ .

$\frac{- 2}{3}$

Step 2.5.3

Move the negative in front of the fraction.

$- \frac{2}{3}$

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 $- \frac{2}{3}$ and a given point $(- 1, \frac{1}{6})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{1}{6}) = - \frac{2}{3} \cdot (x - (- 1))$

Step 3.2

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

$y - \frac{1}{6} = - \frac{2}{3} \cdot (x + 1)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- \frac{2}{3} \cdot (x + 1)$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{1}{6} = 0 + 0 - \frac{2}{3} \cdot (x + 1)$

Step 3.3.1.2

Simplify terms.

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Step 3.3.1.2.1

Apply the distributive property.

$y - \frac{1}{6} = - \frac{2}{3} x - \frac{2}{3} \cdot 1$

Step 3.3.1.2.2

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

$y - \frac{1}{6} = - \frac{x \cdot 2}{3} - \frac{2}{3} \cdot 1$

Step 3.3.1.2.3

Multiply $- 1$ by $1$ .

$y - \frac{1}{6} = - \frac{x \cdot 2}{3} - \frac{2}{3}$

Step 3.3.1.3

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

$y - \frac{1}{6} = - \frac{2 x}{3} - \frac{2}{3}$

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 $\frac{1}{6}$ to both sides of the equation.

$y = - \frac{2 x}{3} - \frac{2}{3} + \frac{1}{6}$

Step 3.3.2.2

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{2 x}{3} - \frac{2}{3} \cdot \frac{2}{2} + \frac{1}{6}$

Step 3.3.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.3.1

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$y = - \frac{2 x}{3} - \frac{2 \cdot 2}{3 \cdot 2} + \frac{1}{6}$

Step 3.3.2.3.2

Multiply $3$ by $2$ .

$y = - \frac{2 x}{3} - \frac{2 \cdot 2}{6} + \frac{1}{6}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{2 x}{3} + \frac{- 2 \cdot 2 + 1}{6}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $- 2$ by $2$ .

$y = - \frac{2 x}{3} + \frac{- 4 + 1}{6}$

Step 3.3.2.5.2

Add $- 4$ and $1$ .

$y = - \frac{2 x}{3} + \frac{- 3}{6}$

Step 3.3.2.6

Simplify each term.

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Step 3.3.2.6.1

Cancel the common factor of $- 3$ and $6$ .

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Step 3.3.2.6.1.1

Factor $3$ out of $- 3$ .

$y = - \frac{2 x}{3} + \frac{3 (- 1)}{6}$

Step 3.3.2.6.1.2

Cancel the common factors.

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Step 3.3.2.6.1.2.1

Factor $3$ out of $6$ .

$y = - \frac{2 x}{3} + \frac{3 \cdot - 1}{3 \cdot 2}$

Step 3.3.2.6.1.2.2

Cancel the common factor.

$y = - \frac{2 x}{3} + \frac{3 \cdot - 1}{3 \cdot 2}$

Step 3.3.2.6.1.2.3

Rewrite the expression.

$y = - \frac{2 x}{3} + \frac{- 1}{2}$

Step 3.3.2.6.2

Move the negative in front of the fraction.

$y = - \frac{2 x}{3} - \frac{1}{2}$

Step 3.3.3

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

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{2}{3} x) - \frac{1}{2}$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{2}{3} x - \frac{1}{2}$

Step 4