Calculus Examples

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

Step 1

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

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Step 1.1

Substitute $2$ in for $x$ .

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

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

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

Step 1.2.2

Remove parentheses.

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

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

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

$y = \frac{1}{{(4 - 1)}^{3}}$

Step 1.2.3.2

Subtract $1$ from $4$ .

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

Step 1.2.3.3

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

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

Step 2

Find the first derivative and evaluate at $x = 2$ and $y = \frac{1}{27}$ 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 $\frac{1}{{(x^{2} - 1)}^{3}}$ as ${({(x^{2} - 1)}^{3})}^{- 1}$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

Multiply $3$ by $- 1$ .

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

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 $f (x) = x^{- 3}$ and $g (x) = x^{2} - 1$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 1$ .

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

Step 2.2.2

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

$- 3 u^{- 4} \frac{d}{d x} [x^{2} - 1]$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} - 1$ .

$- 3 {(x^{2} - 1)}^{- 4} \frac{d}{d x} [x^{2} - 1]$

Step 2.3

Differentiate.

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Step 2.3.1

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

$- 3 {(x^{2} - 1)}^{- 4} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])$

Step 2.3.2

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

$- 3 {(x^{2} - 1)}^{- 4} (2 x + \frac{d}{d x} [- 1])$

Step 2.3.3

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

$- 3 {(x^{2} - 1)}^{- 4} (2 x + 0)$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Add $2 x$ and $0$ .

$- 3 {(x^{2} - 1)}^{- 4} (2 x)$

Step 2.3.4.2

Multiply $2$ by $- 3$ .

$- 6 {(x^{2} - 1)}^{- 4} x$

Step 2.4

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

$- 6 \frac{1}{{(x^{2} - 1)}^{4}} x$

Step 2.5

Combine terms.

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Step 2.5.1

Combine $- 6$ and $\frac{1}{{(x^{2} - 1)}^{4}}$ .

$\frac{- 6}{{(x^{2} - 1)}^{4}} x$

Step 2.5.2

Move the negative in front of the fraction.

$- \frac{6}{{(x^{2} - 1)}^{4}} x$

Step 2.5.3

Combine $x$ and $\frac{6}{{(x^{2} - 1)}^{4}}$ .

$- \frac{x \cdot 6}{{(x^{2} - 1)}^{4}}$

Step 2.5.4

Move $6$ to the left of $x$ .

$- \frac{6 x}{{(x^{2} - 1)}^{4}}$

Step 2.6

Evaluate the derivative at $x = 2$ .

$- \frac{6 (2)}{{({(2)}^{2} - 1)}^{4}}$

Step 2.7

Simplify.

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Step 2.7.1

Multiply $6$ by $2$ .

$- \frac{12}{{(2^{2} - 1)}^{4}}$

Step 2.7.2

Simplify the denominator.

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Step 2.7.2.1

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

$- \frac{12}{{(4 - 1)}^{4}}$

Step 2.7.2.2

Subtract $1$ from $4$ .

$- \frac{12}{3^{4}}$

Step 2.7.2.3

Raise $3$ to the power of $4$ .

$- \frac{12}{81}$

Step 2.7.3

Cancel the common factor of $12$ and $81$ .

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Step 2.7.3.1

Factor $3$ out of $12$ .

$- \frac{3 (4)}{81}$

Step 2.7.3.2

Cancel the common factors.

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Step 2.7.3.2.1

Factor $3$ out of $81$ .

$- \frac{3 \cdot 4}{3 \cdot 27}$

Step 2.7.3.2.2

Cancel the common factor.

$- \frac{3 \cdot 4}{3 \cdot 27}$

Step 2.7.3.2.3

Rewrite the expression.

$- \frac{4}{27}$

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 $- \frac{4}{27}$ and a given point $(2, \frac{1}{27})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{1}{27}) = - \frac{4}{27} \cdot (x - (2))$

Step 3.2

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

$y - \frac{1}{27} = - \frac{4}{27} \cdot (x - 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- \frac{4}{27} \cdot (x - 2)$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{1}{27} = 0 + 0 - \frac{4}{27} \cdot (x - 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{1}{27} = - \frac{4}{27} \cdot (x - 2)$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{1}{27} = - \frac{4}{27} x - \frac{4}{27} \cdot - 2$

Step 3.3.1.4

Combine $x$ and $\frac{4}{27}$ .

$y - \frac{1}{27} = - \frac{x \cdot 4}{27} - \frac{4}{27} \cdot - 2$

Step 3.3.1.5

Multiply $- \frac{4}{27} \cdot - 2$ .

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Step 3.3.1.5.1

Multiply $- 2$ by $- 1$ .

$y - \frac{1}{27} = - \frac{x \cdot 4}{27} + 2 (\frac{4}{27})$

Step 3.3.1.5.2

Combine $2$ and $\frac{4}{27}$ .

$y - \frac{1}{27} = - \frac{x \cdot 4}{27} + \frac{2 \cdot 4}{27}$

Step 3.3.1.5.3

Multiply $2$ by $4$ .

$y - \frac{1}{27} = - \frac{x \cdot 4}{27} + \frac{8}{27}$

Step 3.3.1.6

Move $4$ to the left of $x$ .

$y - \frac{1}{27} = - \frac{4 x}{27} + \frac{8}{27}$

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 $\frac{1}{27}$ to both sides of the equation.

$y = - \frac{4 x}{27} + \frac{8}{27} + \frac{1}{27}$

Step 3.3.2.2

Combine the numerators over the common denominator.

$y = \frac{- 4 x + 8 + 1}{27}$

Step 3.3.2.3

Add $8$ and $1$ .

$y = \frac{- 4 x + 9}{27}$

Step 3.3.2.4

Split the fraction $\frac{- 4 x + 9}{27}$ into two fractions.

$y = \frac{- 4 x}{27} + \frac{9}{27}$

Step 3.3.2.5

Simplify each term.

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Step 3.3.2.5.1

Move the negative in front of the fraction.

$y = - \frac{4 x}{27} + \frac{9}{27}$

Step 3.3.2.5.2

Cancel the common factor of $9$ and $27$ .

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Step 3.3.2.5.2.1

Factor $9$ out of $9$ .

$y = - \frac{4 x}{27} + \frac{9 (1)}{27}$

Step 3.3.2.5.2.2

Cancel the common factors.

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Step 3.3.2.5.2.2.1

Factor $9$ out of $27$ .

$y = - \frac{4 x}{27} + \frac{9 \cdot 1}{9 \cdot 3}$

Step 3.3.2.5.2.2.2

Cancel the common factor.

$y = - \frac{4 x}{27} + \frac{9 \cdot 1}{9 \cdot 3}$

Step 3.3.2.5.2.2.3

Rewrite the expression.

$y = - \frac{4 x}{27} + \frac{1}{3}$

Step 3.3.3

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

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{4}{27} x) + \frac{1}{3}$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{4}{27} x + \frac{1}{3}$

Step 4