Calculus Examples

$f (x) = \frac{x^{7}}{7} - \frac{7}{x^{7}}$ at $x = - 1$

Step 1

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

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Step 1.1

Substitute $- 1$ in for $x$ .

$y = \frac{{(- 1)}^{7}}{7} - \frac{7}{{(- 1)}^{7}}$

Step 1.2

Simplify $\frac{{(- 1)}^{7}}{7} - \frac{7}{{(- 1)}^{7}}$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Raise $- 1$ to the power of $7$ .

$y = \frac{- 1}{7} - \frac{7}{{(- 1)}^{7}}$

Step 1.2.1.2

Move the negative in front of the fraction.

$y = - \frac{1}{7} - \frac{7}{{(- 1)}^{7}}$

Step 1.2.1.3

Raise $- 1$ to the power of $7$ .

$y = - \frac{1}{7} - \frac{7}{- 1}$

Step 1.2.1.4

Divide $7$ by $- 1$ .

$y = - \frac{1}{7} - - 7$

Step 1.2.1.5

Multiply $- 1$ by $- 7$ .

$y = - \frac{1}{7} + 7$

Step 1.2.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = - \frac{1}{7} + 7 \cdot \frac{7}{7}$

Step 1.2.3

Combine $7$ and $\frac{7}{7}$ .

$y = - \frac{1}{7} + \frac{7 \cdot 7}{7}$

Step 1.2.4

Combine the numerators over the common denominator.

$y = \frac{- 1 + 7 \cdot 7}{7}$

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $7$ by $7$ .

$y = \frac{- 1 + 49}{7}$

Step 1.2.5.2

Add $- 1$ and $49$ .

$y = \frac{48}{7}$

Step 2

Find the first derivative and evaluate at $x = - 1$ and $y = \frac{48}{7}$ to find the slope of the tangent line.

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Step 2.1

By the Sum Rule, the derivative of $\frac{x^{7}}{7} - \frac{7}{x^{7}}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{7}}{7}] + \frac{d}{d x} [- \frac{7}{x^{7}}]$ .

$\frac{d}{d x} [\frac{x^{7}}{7}] + \frac{d}{d x} [- \frac{7}{x^{7}}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{x^{7}}{7}]$ .

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Step 2.2.1

Since $\frac{1}{7}$ is constant with respect to $x$ , the derivative of $\frac{x^{7}}{7}$ with respect to $x$ is $\frac{1}{7} \frac{d}{d x} [x^{7}]$ .

$\frac{1}{7} \frac{d}{d x} [x^{7}] + \frac{d}{d x} [- \frac{7}{x^{7}}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$\frac{1}{7} (7 x^{6}) + \frac{d}{d x} [- \frac{7}{x^{7}}]$

Step 2.2.3

Combine $7$ and $\frac{1}{7}$ .

$\frac{7}{7} x^{6} + \frac{d}{d x} [- \frac{7}{x^{7}}]$

Step 2.2.4

Combine $\frac{7}{7}$ and $x^{6}$ .

$\frac{7 x^{6}}{7} + \frac{d}{d x} [- \frac{7}{x^{7}}]$

Step 2.2.5

Cancel the common factor of $7$ .

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Step 2.2.5.1

Cancel the common factor.

$\frac{7 x^{6}}{7} + \frac{d}{d x} [- \frac{7}{x^{7}}]$

Step 2.2.5.2

Divide $x^{6}$ by $1$ .

$x^{6} + \frac{d}{d x} [- \frac{7}{x^{7}}]$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{7}{x^{7}}]$ .

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Step 2.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- \frac{7}{x^{7}}$ with respect to $x$ is $- 7 \frac{d}{d x} [\frac{1}{x^{7}}]$ .

$x^{6} - 7 \frac{d}{d x} [\frac{1}{x^{7}}]$

Step 2.3.2

Rewrite $\frac{1}{x^{7}}$ as ${(x^{7})}^{- 1}$ .

$x^{6} - 7 \frac{d}{d x} [{(x^{7})}^{- 1}]$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{7}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{7}$ .

$x^{6} - 7 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{7}])$

Step 2.3.3.2

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

$x^{6} - 7 (- u^{- 2} \frac{d}{d x} [x^{7}])$

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{7}$ .

$x^{6} - 7 (- {(x^{7})}^{- 2} \frac{d}{d x} [x^{7}])$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$x^{6} - 7 (- {(x^{7})}^{- 2} (7 x^{6}))$

Step 2.3.5

Multiply the exponents in ${(x^{7})}^{- 2}$ .

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Step 2.3.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{6} - 7 (- x^{7 \cdot - 2} (7 x^{6}))$

Step 2.3.5.2

Multiply $7$ by $- 2$ .

$x^{6} - 7 (- x^{- 14} (7 x^{6}))$

Step 2.3.6

Multiply $7$ by $- 1$ .

$x^{6} - 7 (- 7 x^{- 14} x^{6})$

Step 2.3.7

Multiply $x^{- 14}$ by $x^{6}$ by adding the exponents.

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Step 2.3.7.1

Move $x^{6}$ .

$x^{6} - 7 (- 7 (x^{6} x^{- 14}))$

Step 2.3.7.2

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

$x^{6} - 7 (- 7 x^{6 - 14})$

Step 2.3.7.3

Subtract $14$ from $6$ .

$x^{6} - 7 (- 7 x^{- 8})$

Step 2.3.8

Multiply $- 7$ by $- 7$ .

$x^{6} + 49 x^{- 8}$

Step 2.4

Simplify.

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Step 2.4.1

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

$x^{6} + 49 \frac{1}{x^{8}}$

Step 2.4.2

Combine $49$ and $\frac{1}{x^{8}}$ .

$x^{6} + \frac{49}{x^{8}}$

Step 2.5

Evaluate the derivative at $x = - 1$ .

${(- 1)}^{6} + \frac{49}{{(- 1)}^{8}}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

Raise $- 1$ to the power of $6$ .

$1 + \frac{49}{{(- 1)}^{8}}$

Step 2.6.1.2

Raise $- 1$ to the power of $8$ .

$1 + \frac{49}{1}$

Step 2.6.1.3

Divide $49$ by $1$ .

$1 + 49$

Step 2.6.2

Add $1$ and $49$ .

$50$

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 $50$ and a given point $(- 1, \frac{48}{7})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{48}{7}) = 50 \cdot (x - (- 1))$

Step 3.2

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

$y - \frac{48}{7} = 50 \cdot (x + 1)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $50 \cdot (x + 1)$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{48}{7} = 0 + 0 + 50 \cdot (x + 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{48}{7} = 50 \cdot (x + 1)$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{48}{7} = 50 x + 50 \cdot 1$

Step 3.3.1.4

Multiply $50$ by $1$ .

$y - \frac{48}{7} = 50 x + 50$

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 $\frac{48}{7}$ to both sides of the equation.

$y = 50 x + 50 + \frac{48}{7}$

Step 3.3.2.2

To write $50$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = 50 x + 50 \cdot \frac{7}{7} + \frac{48}{7}$

Step 3.3.2.3

Combine $50$ and $\frac{7}{7}$ .

$y = 50 x + \frac{50 \cdot 7}{7} + \frac{48}{7}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = 50 x + \frac{50 \cdot 7 + 48}{7}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $50$ by $7$ .

$y = 50 x + \frac{350 + 48}{7}$

Step 3.3.2.5.2

Add $350$ and $48$ .

$y = 50 x + \frac{398}{7}$

Step 4