Calculus Examples

$y = \frac{3 x^{4} + 9}{x^{2}}$

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 $f (x) = 3 x^{4} + 9$ and $g (x) = x^{2}$ .

$\frac{x^{2} \frac{d}{d x} [3 x^{4} + 9] - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Multiply the exponents in ${(x^{2})}^{2}$ .

Tap for more steps...

Step 2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{2} \frac{d}{d x} [3 x^{4} + 9] - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 2.1.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [3 x^{4} + 9] - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2

By the Sum Rule, the derivative of $3 x^{4} + 9$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [9]$ .

$\frac{x^{2} (\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [9]) - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$\frac{x^{2} (3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [9]) - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{x^{2} (3 (4 x^{3}) + \frac{d}{d x} [9]) - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5

Multiply $4$ by $3$ .

$\frac{x^{2} (12 x^{3} + \frac{d}{d x} [9]) - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{x^{2} (12 x^{3} + 0) - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.7

Add $12 x^{3}$ and $0$ .

$\frac{x^{2} (12 x^{3}) - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 3.1

Move $x^{3}$ .

$\frac{x^{3} x^{2} \cdot 12 - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.2

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

$\frac{x^{3 + 2} \cdot 12 - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.3

Add $3$ and $2$ .

$\frac{x^{5} \cdot 12 - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4

Move $12$ to the left of $x^{5}$ .

$\frac{12 \cdot x^{5} - (3 x^{4} + 9) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5

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

$\frac{12 x^{5} - (3 x^{4} + 9) (2 x)}{x^{4}}$

Step 6

Multiply $2$ by $- 1$ .

$\frac{12 x^{5} - 2 (3 x^{4} + 9) x}{x^{4}}$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Apply the distributive property.

$\frac{12 x^{5} + (- 2 (3 x^{4}) - 2 \cdot 9) x}{x^{4}}$

Step 7.2

Apply the distributive property.

$\frac{12 x^{5} - 2 (3 x^{4}) x - 2 \cdot 9 x}{x^{4}}$

Step 7.3

Simplify the numerator.

Tap for more steps...

Step 7.3.1

Simplify each term.

Tap for more steps...

Step 7.3.1.1

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

Tap for more steps...

Step 7.3.1.1.1

Move $x$ .

$\frac{12 x^{5} - 2 (3 (x \cdot x^{4})) - 2 \cdot 9 x}{x^{4}}$

Step 7.3.1.1.2

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

Tap for more steps...

Step 7.3.1.1.2.1

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

$\frac{12 x^{5} - 2 (3 (x^{1} x^{4})) - 2 \cdot 9 x}{x^{4}}$

Step 7.3.1.1.2.2

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

$\frac{12 x^{5} - 2 (3 x^{1 + 4}) - 2 \cdot 9 x}{x^{4}}$

Step 7.3.1.1.3

Add $1$ and $4$ .

$\frac{12 x^{5} - 2 (3 x^{5}) - 2 \cdot 9 x}{x^{4}}$

Step 7.3.1.2

Multiply $3$ by $- 2$ .

$\frac{12 x^{5} - 6 x^{5} - 2 \cdot 9 x}{x^{4}}$

Step 7.3.1.3

Multiply $- 2$ by $9$ .

$\frac{12 x^{5} - 6 x^{5} - 18 x}{x^{4}}$

Step 7.3.2

Subtract $6 x^{5}$ from $12 x^{5}$ .

$\frac{6 x^{5} - 18 x}{x^{4}}$

Step 7.4

Factor $6 x$ out of $6 x^{5} - 18 x$ .

Tap for more steps...

Step 7.4.1

Factor $6 x$ out of $6 x^{5}$ .

$\frac{6 x (x^{4}) - 18 x}{x^{4}}$

Step 7.4.2

Factor $6 x$ out of $- 18 x$ .

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

Step 7.4.3

Factor $6 x$ out of $6 x (x^{4}) + 6 x (- 3)$ .

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

Step 7.5

Cancel the common factor of $x$ and $x^{4}$ .

Tap for more steps...

Step 7.5.1

Factor $x$ out of $6 x (x^{4} - 3)$ .

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

Step 7.5.2

Cancel the common factors.

Tap for more steps...

Step 7.5.2.1

Factor $x$ out of $x^{4}$ .

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

Step 7.5.2.2

Cancel the common factor.

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

Step 7.5.2.3

Rewrite the expression.

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