Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Multiply $3$ by $2$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $2 x$ and $0$ .

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

Step 3

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

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Step 3.1

Move $x$ .

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

Step 3.2

Multiply $x$ by $x^{3}$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

Add $1$ and $3$ .

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

Step 4

Move $2$ to the left of $x^{4}$ .

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

Step 5

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

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

Step 6

Simplify with factoring out.

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Step 6.1

Multiply $3$ by $- 1$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 7

Cancel the common factors.

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Simplify the numerator.

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Step 8.2.1

Multiply $- 3$ by $- 1$ .

$\frac{2 x^{2} - 3 x^{2} + 3}{x^{4}}$

Step 8.2.2

Subtract $3 x^{2}$ from $2 x^{2}$ .

$\frac{- x^{2} + 3}{x^{4}}$

Step 8.3

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

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

Step 8.4

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 8.5

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

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

Step 8.6

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

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

Step 8.7

Move the negative in front of the fraction.

$- \frac{x^{2} - 3}{x^{4}}$