Calculus Examples

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

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 $f (x) = x^{4}$ and $g (x) = 1 - x^{3}$ .

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

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 $n = 4$ .

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

Step 2.2

Move $4$ to the left of $1 - x^{3}$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $0$ and $\frac{d}{d x} [- x^{3}]$ .

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

Step 2.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

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

Step 2.7

Multiply.

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Step 2.7.1

Multiply $- 1$ by $- 1$ .

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

Step 2.7.2

Multiply $x^{4}$ by $1$ .

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

Step 2.8

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

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

Step 3

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 3.1

Move $x^{2}$ .

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

Step 3.2

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

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

Step 3.3

Add $2$ and $4$ .

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

Step 4

Move $3$ to the left of $x^{6}$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Simplify the numerator.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $4$ by $1$ .

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

Step 5.3.1.2

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

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Step 5.3.1.2.1

Move $x^{3}$ .

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

Step 5.3.1.2.2

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

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

Step 5.3.1.2.3

Add $3$ and $3$ .

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

Step 5.3.1.3

Multiply $- 1$ by $4$ .

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

Step 5.3.2

Add $- 4 x^{6}$ and $3 x^{6}$ .

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

Step 5.4

Reorder terms.

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