Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Move $2$ to the left of $x^{3} - 1$ .

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

Step 2.3

By the Sum Rule, the derivative of $x^{3} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 1]$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $3 x^{2}$ and $0$ .

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

Step 2.6.2

Multiply $3$ by $- 1$ .

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

Step 3

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

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Step 3.1

Move $x^{2}$ .

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

Step 3.2

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

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

Step 3.3

Add $2$ and $2$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

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

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Step 4.3.1.1.1

Move $x$ .

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

Step 4.3.1.1.2

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

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Step 4.3.1.1.2.1

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

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

Step 4.3.1.1.2.2

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

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

Step 4.3.1.1.3

Add $1$ and $3$ .

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

Step 4.3.1.2

Multiply $2$ by $- 1$ .

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

Step 4.3.2

Subtract $3 x^{4}$ from $2 x^{4}$ .

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