Calculus Examples

$y = \frac{x^{9}}{x^{4}}$

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

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

Step 2

Differentiate using the Power Rule.

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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 $n = 9$ .

$\frac{x^{4} (9 x^{8}) - x^{9} \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 2.2

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

$\frac{x^{4} (9 x^{8}) - x^{9} (4 x^{3})}{{(x^{4})}^{2}}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Rewrite using the commutative property of multiplication.

$\frac{9 x^{4} x^{8} - x^{9} (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.1.1.2

Multiply $x^{4}$ by $x^{8}$ by adding the exponents.

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Step 3.1.1.2.1

Move $x^{8}$ .

$\frac{9 (x^{8} x^{4}) - x^{9} (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.1.1.2.2

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

$\frac{9 x^{8 + 4} - x^{9} (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.1.1.2.3

Add $8$ and $4$ .

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

Step 3.1.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.1.1.4

Multiply $x^{9}$ by $x^{3}$ by adding the exponents.

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Step 3.1.1.4.1

Move $x^{3}$ .

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

Step 3.1.1.4.2

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

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

Step 3.1.1.4.3

Add $3$ and $9$ .

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

Step 3.1.1.5

Multiply $- 1$ by $4$ .

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

Step 3.1.2

Subtract $4 x^{12}$ from $9 x^{12}$ .

$\frac{5 x^{12}}{{(x^{4})}^{2}}$

Step 3.2

Combine terms.

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Step 3.2.1

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

Multiply $4$ by $2$ .

$\frac{5 x^{12}}{x^{8}}$

Step 3.2.2

Cancel the common factor of $x^{12}$ and $x^{8}$ .

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Step 3.2.2.1

Factor $x^{8}$ out of $5 x^{12}$ .

$\frac{x^{8} (5 x^{4})}{x^{8}}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

Multiply by $1$ .

$\frac{x^{8} (5 x^{4})}{x^{8} \cdot 1}$

Step 3.2.2.2.2

Cancel the common factor.

$\frac{x^{8} (5 x^{4})}{x^{8} \cdot 1}$

Step 3.2.2.2.3

Rewrite the expression.

$\frac{5 x^{4}}{1}$

Step 3.2.2.2.4

Divide $5 x^{4}$ by $1$ .

$5 x^{4}$