Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

Move $3$ to the left of $x - 1$ .

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Add $1$ and $0$ .

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

Step 3.2.6.2

Multiply $- 1$ by $1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Simplify the numerator.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

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

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Step 3.3.3.1.1.1

Move $x^{2}$ .

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

Step 3.3.3.1.1.2

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

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Step 3.3.3.1.1.2.1

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

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

Step 3.3.3.1.1.2.2

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

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

Step 3.3.3.1.1.3

Add $2$ and $1$ .

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

Step 3.3.3.1.2

Multiply $3$ by $- 1$ .

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

Step 3.3.3.2

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

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

Step 3.3.4

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

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Step 3.3.4.1

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

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

Step 3.3.4.2

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

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

Step 3.3.4.3

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

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

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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