Calculus Examples

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

Step 1

Find the corresponding $y$ -value to $x = 2$ .

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Step 1.1

Substitute $2$ in for $x$ .

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

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

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

Step 1.2.2

Remove parentheses.

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

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

Raise $2$ to the power of $3$ .

$y = \frac{1}{{(8 - (2))}^{2}}$

Step 1.2.3.2

Multiply $- 1$ by $2$ .

$y = \frac{1}{{(8 - 2)}^{2}}$

Step 1.2.3.3

Subtract $2$ from $8$ .

$y = \frac{1}{6^{2}}$

Step 1.2.3.4

Raise $6$ to the power of $2$ .

$y = \frac{1}{36}$

Step 2

Find the first derivative and evaluate at $x = 2$ and $y = \frac{1}{36}$ to find the slope of the tangent line.

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Step 2.1

Apply basic rules of exponents.

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Step 2.1.1

Rewrite $\frac{1}{{(x^{3} - x)}^{2}}$ as ${({(x^{3} - x)}^{2})}^{- 1}$ .

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

Step 2.1.2

Multiply the exponents in ${({(x^{3} - x)}^{2})}^{- 1}$ .

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Step 2.1.2.1

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

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

Step 2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = x^{3} - x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{3} - x$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $x^{3} - x$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

$- 2 {(x^{3} - x)}^{- 3} (3 x^{2} - 1 \cdot 1)$

Step 2.3.5

Multiply $- 1$ by $1$ .

$- 2 {(x^{3} - x)}^{- 3} (3 x^{2} - 1)$

Step 2.4

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

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

Step 2.5

Simplify.

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Step 2.5.1

Combine terms.

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Step 2.5.1.1

Combine $- 2$ and $\frac{1}{{(x^{3} - x)}^{3}}$ .

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

Step 2.5.1.2

Move the negative in front of the fraction.

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

Step 2.5.2

Reorder the factors of $- \frac{2}{{(x^{3} - x)}^{3}} (3 x^{2} - 1)$ .

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

Step 2.6

Evaluate the derivative at $x = 2$ .

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

Step 2.7

Simplify.

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Step 2.7.1

Simplify the denominator.

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Step 2.7.1.1

Raise $2$ to the power of $3$ .

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

Step 2.7.1.2

Multiply $- 1$ by $2$ .

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

Step 2.7.1.3

Subtract $2$ from $8$ .

$- (3 {(2)}^{2} - 1) \frac{2}{6^{3}}$

Step 2.7.1.4

Raise $6$ to the power of $3$ .

$- (3 {(2)}^{2} - 1) \frac{2}{216}$

Step 2.7.2

Simplify terms.

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Step 2.7.2.1

Simplify each term.

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Step 2.7.2.1.1

Raise $2$ to the power of $2$ .

$- (3 \cdot 4 - 1) \frac{2}{216}$

Step 2.7.2.1.2

Multiply $3$ by $4$ .

$- (12 - 1) \frac{2}{216}$

Step 2.7.2.2

Simplify terms.

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Step 2.7.2.2.1

Subtract $1$ from $12$ .

$- 1 \cdot 11 \frac{2}{216}$

Step 2.7.2.2.2

Multiply $- 1$ by $11$ .

$- 11 (\frac{2}{216})$

Step 2.7.2.2.3

Cancel the common factor of $2$ and $216$ .

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Step 2.7.2.2.3.1

Factor $2$ out of $2$ .

$- 11 \frac{2 (1)}{216}$

Step 2.7.2.2.3.2

Cancel the common factors.

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Step 2.7.2.2.3.2.1

Factor $2$ out of $216$ .

$- 11 \frac{2 \cdot 1}{2 \cdot 108}$

Step 2.7.2.2.3.2.2

Cancel the common factor.

$- 11 \frac{2 \cdot 1}{2 \cdot 108}$

Step 2.7.2.2.3.2.3

Rewrite the expression.

$- 11 (\frac{1}{108})$

Step 2.7.2.2.4

Combine $- 11$ and $\frac{1}{108}$ .

$\frac{- 11}{108}$

Step 2.7.2.2.5

Move the negative in front of the fraction.

$- \frac{11}{108}$

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 $- \frac{11}{108}$ and a given point $(2, \frac{1}{36})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{1}{36}) = - \frac{11}{108} \cdot (x - (2))$

Step 3.2

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

$y - \frac{1}{36} = - \frac{11}{108} \cdot (x - 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- \frac{11}{108} \cdot (x - 2)$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{1}{36} = 0 + 0 - \frac{11}{108} \cdot (x - 2)$

Step 3.3.1.2

Simplify terms.

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Step 3.3.1.2.1

Apply the distributive property.

$y - \frac{1}{36} = - \frac{11}{108} x - \frac{11}{108} \cdot - 2$

Step 3.3.1.2.2

Combine $x$ and $\frac{11}{108}$ .

$y - \frac{1}{36} = - \frac{x \cdot 11}{108} - \frac{11}{108} \cdot - 2$

Step 3.3.1.2.3

Cancel the common factor of $2$ .

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Step 3.3.1.2.3.1

Move the leading negative in $- \frac{11}{108}$ into the numerator.

$y - \frac{1}{36} = - \frac{x \cdot 11}{108} + \frac{- 11}{108} \cdot - 2$

Step 3.3.1.2.3.2

Factor $2$ out of $108$ .

$y - \frac{1}{36} = - \frac{x \cdot 11}{108} + \frac{- 11}{2 (54)} \cdot - 2$

Step 3.3.1.2.3.3

Factor $2$ out of $- 2$ .

$y - \frac{1}{36} = - \frac{x \cdot 11}{108} + \frac{- 11}{2 \cdot 54} \cdot (2 \cdot - 1)$

Step 3.3.1.2.3.4

Cancel the common factor.

$y - \frac{1}{36} = - \frac{x \cdot 11}{108} + \frac{- 11}{2 \cdot 54} \cdot (2 \cdot - 1)$

Step 3.3.1.2.3.5

Rewrite the expression.

$y - \frac{1}{36} = - \frac{x \cdot 11}{108} + \frac{- 11}{54} \cdot - 1$

Step 3.3.1.2.4

Combine $\frac{- 11}{54}$ and $- 1$ .

$y - \frac{1}{36} = - \frac{x \cdot 11}{108} + \frac{- 11 \cdot - 1}{54}$

Step 3.3.1.2.5

Multiply $- 11$ by $- 1$ .

$y - \frac{1}{36} = - \frac{x \cdot 11}{108} + \frac{11}{54}$

Step 3.3.1.3

Move $11$ to the left of $x$ .

$y - \frac{1}{36} = - \frac{11 x}{108} + \frac{11}{54}$

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 $\frac{1}{36}$ to both sides of the equation.

$y = - \frac{11 x}{108} + \frac{11}{54} + \frac{1}{36}$

Step 3.3.2.2

To write $\frac{11}{54}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{11 x}{108} + \frac{11}{54} \cdot \frac{2}{2} + \frac{1}{36}$

Step 3.3.2.3

To write $\frac{1}{36}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{11 x}{108} + \frac{11}{54} \cdot \frac{2}{2} + \frac{1}{36} \cdot \frac{3}{3}$

Step 3.3.2.4

Write each expression with a common denominator of $108$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.4.1

Multiply $\frac{11}{54}$ by $\frac{2}{2}$ .

$y = - \frac{11 x}{108} + \frac{11 \cdot 2}{54 \cdot 2} + \frac{1}{36} \cdot \frac{3}{3}$

Step 3.3.2.4.2

Multiply $54$ by $2$ .

$y = - \frac{11 x}{108} + \frac{11 \cdot 2}{108} + \frac{1}{36} \cdot \frac{3}{3}$

Step 3.3.2.4.3

Multiply $\frac{1}{36}$ by $\frac{3}{3}$ .

$y = - \frac{11 x}{108} + \frac{11 \cdot 2}{108} + \frac{3}{36 \cdot 3}$

Step 3.3.2.4.4

Multiply $36$ by $3$ .

$y = - \frac{11 x}{108} + \frac{11 \cdot 2}{108} + \frac{3}{108}$

Step 3.3.2.5

Combine the numerators over the common denominator.

$y = - \frac{11 x}{108} + \frac{11 \cdot 2 + 3}{108}$

Step 3.3.2.6

Simplify the numerator.

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Step 3.3.2.6.1

Multiply $11$ by $2$ .

$y = - \frac{11 x}{108} + \frac{22 + 3}{108}$

Step 3.3.2.6.2

Add $22$ and $3$ .

$y = - \frac{11 x}{108} + \frac{25}{108}$

Step 3.3.3

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

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{11}{108} x) + \frac{25}{108}$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{11}{108} x + \frac{25}{108}$

Step 4