Calculus Examples

$y = \frac{9 x^{2} - 1}{3 x - 1}$

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 $f (x) = 9 x^{2} - 1$ and $g (x) = 3 x - 1$ .

$\frac{(3 x - 1) \frac{d}{d x} [9 x^{2} - 1] - (9 x^{2} - 1) \frac{d}{d x} [3 x - 1]}{{(3 x - 1)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $9 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 1]$ .

$\frac{(3 x - 1) (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 1]) - (9 x^{2} - 1) \frac{d}{d x} [3 x - 1]}{{(3 x - 1)}^{2}}$

Step 2.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$\frac{(3 x - 1) (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]) - (9 x^{2} - 1) \frac{d}{d x} [3 x - 1]}{{(3 x - 1)}^{2}}$

Step 2.3

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

$\frac{(3 x - 1) (9 (2 x) + \frac{d}{d x} [- 1]) - (9 x^{2} - 1) \frac{d}{d x} [3 x - 1]}{{(3 x - 1)}^{2}}$

Step 2.4

Multiply $2$ by $9$ .

$\frac{(3 x - 1) (18 x + \frac{d}{d x} [- 1]) - (9 x^{2} - 1) \frac{d}{d x} [3 x - 1]}{{(3 x - 1)}^{2}}$

Step 2.5

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

$\frac{(3 x - 1) (18 x + 0) - (9 x^{2} - 1) \frac{d}{d x} [3 x - 1]}{{(3 x - 1)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $18 x$ and $0$ .

$\frac{(3 x - 1) (18 x) - (9 x^{2} - 1) \frac{d}{d x} [3 x - 1]}{{(3 x - 1)}^{2}}$

Step 2.6.2

Move $18$ to the left of $3 x - 1$ .

$\frac{18 \cdot (3 x - 1) x - (9 x^{2} - 1) \frac{d}{d x} [3 x - 1]}{{(3 x - 1)}^{2}}$

Step 2.7

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

$\frac{18 (3 x - 1) x - (9 x^{2} - 1) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1])}{{(3 x - 1)}^{2}}$

Step 2.8

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

$\frac{18 (3 x - 1) x - (9 x^{2} - 1) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])}{{(3 x - 1)}^{2}}$

Step 2.9

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

$\frac{18 (3 x - 1) x - (9 x^{2} - 1) (3 \cdot 1 + \frac{d}{d x} [- 1])}{{(3 x - 1)}^{2}}$

Step 2.10

Multiply $3$ by $1$ .

$\frac{18 (3 x - 1) x - (9 x^{2} - 1) (3 + \frac{d}{d x} [- 1])}{{(3 x - 1)}^{2}}$

Step 2.11

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

$\frac{18 (3 x - 1) x - (9 x^{2} - 1) (3 + 0)}{{(3 x - 1)}^{2}}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $3$ and $0$ .

$\frac{18 (3 x - 1) x - (9 x^{2} - 1) \cdot 3}{{(3 x - 1)}^{2}}$

Step 2.12.2

Multiply $3$ by $- 1$ .

$\frac{18 (3 x - 1) x - 3 (9 x^{2} - 1)}{{(3 x - 1)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(18 (3 x) + 18 \cdot - 1) x - 3 (9 x^{2} - 1)}{{(3 x - 1)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{18 (3 x) x + 18 \cdot - 1 x - 3 (9 x^{2} - 1)}{{(3 x - 1)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{18 (3 x) x + 18 \cdot - 1 x - 3 (9 x^{2}) - 3 \cdot - 1}{{(3 x - 1)}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.1.1.1

Move $x$ .

$\frac{18 (3 (x \cdot x)) + 18 \cdot - 1 x - 3 (9 x^{2}) - 3 \cdot - 1}{{(3 x - 1)}^{2}}$

Step 3.4.1.1.2

Multiply $x$ by $x$ .

$\frac{18 (3 x^{2}) + 18 \cdot - 1 x - 3 (9 x^{2}) - 3 \cdot - 1}{{(3 x - 1)}^{2}}$

Step 3.4.1.2

Multiply $3$ by $18$ .

$\frac{54 x^{2} + 18 \cdot - 1 x - 3 (9 x^{2}) - 3 \cdot - 1}{{(3 x - 1)}^{2}}$

Step 3.4.1.3

Multiply $18$ by $- 1$ .

$\frac{54 x^{2} - 18 x - 3 (9 x^{2}) - 3 \cdot - 1}{{(3 x - 1)}^{2}}$

Step 3.4.1.4

Multiply $9$ by $- 3$ .

$\frac{54 x^{2} - 18 x - 27 x^{2} - 3 \cdot - 1}{{(3 x - 1)}^{2}}$

Step 3.4.1.5

Multiply $- 3$ by $- 1$ .

$\frac{54 x^{2} - 18 x - 27 x^{2} + 3}{{(3 x - 1)}^{2}}$

Step 3.4.2

Subtract $27 x^{2}$ from $54 x^{2}$ .

$\frac{27 x^{2} - 18 x + 3}{{(3 x - 1)}^{2}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $3$ out of $27 x^{2} - 18 x + 3$ .

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Step 3.5.1.1

Factor $3$ out of $27 x^{2}$ .

$\frac{3 (9 x^{2}) - 18 x + 3}{{(3 x - 1)}^{2}}$

Step 3.5.1.2

Factor $3$ out of $- 18 x$ .

$\frac{3 (9 x^{2}) + 3 (- 6 x) + 3}{{(3 x - 1)}^{2}}$

Step 3.5.1.3

Factor $3$ out of $3$ .

$\frac{3 (9 x^{2}) + 3 (- 6 x) + 3 (1)}{{(3 x - 1)}^{2}}$

Step 3.5.1.4

Factor $3$ out of $3 (9 x^{2}) + 3 (- 6 x)$ .

$\frac{3 (9 x^{2} - 6 x) + 3 (1)}{{(3 x - 1)}^{2}}$

Step 3.5.1.5

Factor $3$ out of $3 (9 x^{2} - 6 x) + 3 (1)$ .

$\frac{3 (9 x^{2} - 6 x + 1)}{{(3 x - 1)}^{2}}$

Step 3.5.2

Factor using the perfect square rule.

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Step 3.5.2.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

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

Step 3.5.2.2

Rewrite $1$ as $1^{2}$ .

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

Step 3.5.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot (3 x) \cdot 1$

Step 3.5.2.4

Rewrite the polynomial.

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

Step 3.5.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 x$ and $b = 1$ .

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

Step 3.6

Cancel the common factor of ${(3 x - 1)}^{2}$ .

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Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Divide $3$ by $1$ .

$3$