Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

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 $n = 2$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.2.6

Multiply $3$ by $1$ .

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

Step 3.2.7

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

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

Step 3.2.8

Simplify the expression.

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Step 3.2.8.1

Add $3$ and $0$ .

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

Step 3.2.8.2

Multiply $3$ by $- 1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Apply the distributive property.

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

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 $x$ by adding the exponents.

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Step 3.3.3.1.1.1

Move $x$ .

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

Step 3.3.3.1.1.2

Multiply $x$ by $x$ .

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

Step 3.3.3.1.2

Multiply $3$ by $2$ .

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

Step 3.3.3.1.3

Multiply $2$ by $- 1$ .

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

Step 3.3.3.2

Subtract $3 x^{2}$ from $6 x^{2}$ .

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

Step 3.3.4

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

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Step 3.3.4.1

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

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

Step 3.3.4.2

Factor $x$ out of $- 2 x$ .

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

Step 3.3.4.3

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

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

Step 4

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

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

Step 5

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

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