Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{4} - 3 x^{2}$ and $g (x) = {(x^{2} - 1)}^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.2

By the Sum Rule, the derivative of $x^{4} - 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 3 x^{2}]$ .

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

Step 2.3

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

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

Step 2.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

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

Step 2.5

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

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

Step 2.6

Multiply $2$ by $- 3$ .

$\frac{{(x^{2} - 1)}^{2} (4 x^{3} - 6 x) - (x^{4} - 3 x^{2}) \frac{d}{d x} [{(x^{2} - 1)}^{2}]}{{(x^{2} - 1)}^{4}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} - 1$ .

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Step 3.1

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

$\frac{{(x^{2} - 1)}^{2} (4 x^{3} - 6 x) - (x^{4} - 3 x^{2}) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} - 1])}{{(x^{2} - 1)}^{4}}$

Step 3.2

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

$\frac{{(x^{2} - 1)}^{2} (4 x^{3} - 6 x) - (x^{4} - 3 x^{2}) (2 u \frac{d}{d x} [x^{2} - 1])}{{(x^{2} - 1)}^{4}}$

Step 3.3

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

$\frac{{(x^{2} - 1)}^{2} (4 x^{3} - 6 x) - (x^{4} - 3 x^{2}) (2 (x^{2} - 1) \frac{d}{d x} [x^{2} - 1])}{{(x^{2} - 1)}^{4}}$

Step 4

Simplify with factoring out.

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Step 4.1

Multiply $2$ by $- 1$ .

$\frac{{(x^{2} - 1)}^{2} (4 x^{3} - 6 x) - 2 (x^{4} - 3 x^{2}) ((x^{2} - 1) \frac{d}{d x} [x^{2} - 1])}{{(x^{2} - 1)}^{4}}$

Step 4.2

Factor $x^{2} - 1$ out of ${(x^{2} - 1)}^{2} (4 x^{3} - 6 x) - 2 (x^{4} - 3 x^{2}) ((x^{2} - 1) \frac{d}{d x} [x^{2} - 1])$ .

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Step 4.2.1

Factor $x^{2} - 1$ out of ${(x^{2} - 1)}^{2} (4 x^{3} - 6 x)$ .

$\frac{(x^{2} - 1) ((x^{2} - 1) (4 x^{3} - 6 x)) - 2 (x^{4} - 3 x^{2}) ((x^{2} - 1) \frac{d}{d x} [x^{2} - 1])}{{(x^{2} - 1)}^{4}}$

Step 4.2.2

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

$\frac{(x^{2} - 1) ((x^{2} - 1) (4 x^{3} - 6 x)) + (x^{2} - 1) (- 2 (x^{4} - 3 x^{2}) (\frac{d}{d x} [x^{2} - 1]))}{{(x^{2} - 1)}^{4}}$

Step 4.2.3

Factor $x^{2} - 1$ out of $(x^{2} - 1) ((x^{2} - 1) (4 x^{3} - 6 x)) + (x^{2} - 1) (- 2 (x^{4} - 3 x^{2}) (\frac{d}{d x} [x^{2} - 1]))$ .

$\frac{(x^{2} - 1) ((x^{2} - 1) (4 x^{3} - 6 x) - 2 (x^{4} - 3 x^{2}) (\frac{d}{d x} [x^{2} - 1]))}{{(x^{2} - 1)}^{4}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $x^{2} - 1$ out of ${(x^{2} - 1)}^{4}$ .

$\frac{(x^{2} - 1) ((x^{2} - 1) (4 x^{3} - 6 x) - 2 (x^{4} - 3 x^{2}) (\frac{d}{d x} [x^{2} - 1]))}{(x^{2} - 1) {(x^{2} - 1)}^{3}}$

Step 5.2

Cancel the common factor.

$\frac{(x^{2} - 1) ((x^{2} - 1) (4 x^{3} - 6 x) - 2 (x^{4} - 3 x^{2}) (\frac{d}{d x} [x^{2} - 1]))}{(x^{2} - 1) {(x^{2} - 1)}^{3}}$

Step 5.3

Rewrite the expression.

$\frac{(x^{2} - 1) (4 x^{3} - 6 x) - 2 (x^{4} - 3 x^{2}) (\frac{d}{d x} [x^{2} - 1])}{{(x^{2} - 1)}^{3}}$

Step 6

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

$\frac{(x^{2} - 1) (4 x^{3} - 6 x) - 2 (x^{4} - 3 x^{2}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{3}}$

Step 7

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

$\frac{(x^{2} - 1) (4 x^{3} - 6 x) - 2 (x^{4} - 3 x^{2}) (2 x + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{3}}$

Step 8

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

$\frac{(x^{2} - 1) (4 x^{3} - 6 x) - 2 (x^{4} - 3 x^{2}) (2 x + 0)}{{(x^{2} - 1)}^{3}}$

Step 9

Simplify the expression.

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Step 9.1

Add $2 x$ and $0$ .

$\frac{(x^{2} - 1) (4 x^{3} - 6 x) - 2 (x^{4} - 3 x^{2}) (2 x)}{{(x^{2} - 1)}^{3}}$

Step 9.2

Multiply $2$ by $- 2$ .

$\frac{(x^{2} - 1) (4 x^{3} - 6 x) - 4 (x^{4} - 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{(x^{2} - 1) (4 x^{3} - 6 x) + (- 4 x^{4} - 4 (- 3 x^{2})) x}{{(x^{2} - 1)}^{3}}$

Step 10.2

Apply the distributive property.

$\frac{(x^{2} - 1) (4 x^{3} - 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

Simplify each term.

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Step 10.3.1.1

Expand $(x^{2} - 1) (4 x^{3} - 6 x)$ using the FOIL Method.

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Step 10.3.1.1.1

Apply the distributive property.

$\frac{x^{2} (4 x^{3} - 6 x) - 1 (4 x^{3} - 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.1.2

Apply the distributive property.

$\frac{x^{2} (4 x^{3}) + x^{2} (- 6 x) - 1 (4 x^{3} - 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.1.3

Apply the distributive property.

$\frac{x^{2} (4 x^{3}) + x^{2} (- 6 x) - 1 (4 x^{3}) - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2

Simplify and combine like terms.

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Step 10.3.1.2.1

Simplify each term.

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Step 10.3.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 x^{2} x^{3} + x^{2} (- 6 x) - 1 (4 x^{3}) - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.1.2

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

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Step 10.3.1.2.1.2.1

Move $x^{3}$ .

$\frac{4 (x^{3} x^{2}) + x^{2} (- 6 x) - 1 (4 x^{3}) - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.1.2.2

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

$\frac{4 x^{3 + 2} + x^{2} (- 6 x) - 1 (4 x^{3}) - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.1.2.3

Add $3$ and $2$ .

$\frac{4 x^{5} + x^{2} (- 6 x) - 1 (4 x^{3}) - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{4 x^{5} - 6 x^{2} x - 1 (4 x^{3}) - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.1.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 10.3.1.2.1.4.1

Move $x$ .

$\frac{4 x^{5} - 6 (x \cdot x^{2}) - 1 (4 x^{3}) - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.1.4.2

Multiply $x$ by $x^{2}$ .

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Step 10.3.1.2.1.4.2.1

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

$\frac{4 x^{5} - 6 (x^{1} x^{2}) - 1 (4 x^{3}) - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.1.4.2.2

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

$\frac{4 x^{5} - 6 x^{1 + 2} - 1 (4 x^{3}) - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.1.4.3

Add $1$ and $2$ .

$\frac{4 x^{5} - 6 x^{3} - 1 (4 x^{3}) - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.1.5

Multiply $4$ by $- 1$ .

$\frac{4 x^{5} - 6 x^{3} - 4 x^{3} - 1 (- 6 x) - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.1.6

Multiply $- 6$ by $- 1$ .

$\frac{4 x^{5} - 6 x^{3} - 4 x^{3} + 6 x - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.2.2

Subtract $4 x^{3}$ from $- 6 x^{3}$ .

$\frac{4 x^{5} - 10 x^{3} + 6 x - 4 x^{4} x - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.3

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 10.3.1.3.1

Move $x$ .

$\frac{4 x^{5} - 10 x^{3} + 6 x - 4 (x \cdot x^{4}) - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.3.2

Multiply $x$ by $x^{4}$ .

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Step 10.3.1.3.2.1

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

$\frac{4 x^{5} - 10 x^{3} + 6 x - 4 (x^{1} x^{4}) - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.3.2.2

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

$\frac{4 x^{5} - 10 x^{3} + 6 x - 4 x^{1 + 4} - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.3.3

Add $1$ and $4$ .

$\frac{4 x^{5} - 10 x^{3} + 6 x - 4 x^{5} - 4 (- 3 x^{2}) x}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 10.3.1.4.1

Move $x$ .

$\frac{4 x^{5} - 10 x^{3} + 6 x - 4 x^{5} - 4 (- 3 (x \cdot x^{2}))}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.4.2

Multiply $x$ by $x^{2}$ .

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Step 10.3.1.4.2.1

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

$\frac{4 x^{5} - 10 x^{3} + 6 x - 4 x^{5} - 4 (- 3 (x^{1} x^{2}))}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.4.2.2

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

$\frac{4 x^{5} - 10 x^{3} + 6 x - 4 x^{5} - 4 (- 3 x^{1 + 2})}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.4.3

Add $1$ and $2$ .

$\frac{4 x^{5} - 10 x^{3} + 6 x - 4 x^{5} - 4 (- 3 x^{3})}{{(x^{2} - 1)}^{3}}$

Step 10.3.1.5

Multiply $- 3$ by $- 4$ .

$\frac{4 x^{5} - 10 x^{3} + 6 x - 4 x^{5} + 12 x^{3}}{{(x^{2} - 1)}^{3}}$

Step 10.3.2

Combine the opposite terms in $4 x^{5} - 10 x^{3} + 6 x - 4 x^{5} + 12 x^{3}$ .

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Step 10.3.2.1

Subtract $4 x^{5}$ from $4 x^{5}$ .

$\frac{- 10 x^{3} + 6 x + 0 + 12 x^{3}}{{(x^{2} - 1)}^{3}}$

Step 10.3.2.2

Add $- 10 x^{3} + 6 x$ and $0$ .

$\frac{- 10 x^{3} + 6 x + 12 x^{3}}{{(x^{2} - 1)}^{3}}$

Step 10.3.3

Add $- 10 x^{3}$ and $12 x^{3}$ .

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

Step 10.4

Factor $2 x$ out of $2 x^{3} + 6 x$ .

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Step 10.4.1

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

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

Step 10.4.2

Factor $2 x$ out of $6 x$ .

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

Step 10.4.3

Factor $2 x$ out of $2 x (x^{2}) + 2 x (3)$ .

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

Step 10.5

Simplify the denominator.

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Step 10.5.1

Rewrite $1$ as $1^{2}$ .

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

Step 10.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 10.5.3

Apply the product rule to $(x + 1) (x - 1)$ .

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