Calculus Examples

$y = \frac{x^{6}}{x^{4}}$

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 $f (x) = x^{6}$ and $g (x) = x^{4}$ .

$\frac{x^{4} \frac{d}{d x} [x^{6}] - x^{6} \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

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 $n = 6$ .

$\frac{x^{4} (6 x^{5}) - x^{6} \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 2.2

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

$\frac{x^{4} (6 x^{5}) - x^{6} (4 x^{3})}{{(x^{4})}^{2}}$

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.

$\frac{6 x^{4} x^{5} - x^{6} (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.1.1.2

Multiply $x^{4}$ by $x^{5}$ by adding the exponents.

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Step 3.1.1.2.1

Move $x^{5}$ .

$\frac{6 (x^{5} x^{4}) - x^{6} (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.1.1.2.2

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

$\frac{6 x^{5 + 4} - x^{6} (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.1.1.2.3

Add $5$ and $4$ .

$\frac{6 x^{9} - x^{6} (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.1.1.3

Rewrite using the commutative property of multiplication.

$\frac{6 x^{9} - 1 \cdot 4 x^{6} x^{3}}{{(x^{4})}^{2}}$

Step 3.1.1.4

Multiply $x^{6}$ by $x^{3}$ by adding the exponents.

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Step 3.1.1.4.1

Move $x^{3}$ .

$\frac{6 x^{9} - 1 \cdot 4 (x^{3} x^{6})}{{(x^{4})}^{2}}$

Step 3.1.1.4.2

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

$\frac{6 x^{9} - 1 \cdot 4 x^{3 + 6}}{{(x^{4})}^{2}}$

Step 3.1.1.4.3

Add $3$ and $6$ .

$\frac{6 x^{9} - 1 \cdot 4 x^{9}}{{(x^{4})}^{2}}$

Step 3.1.1.5

Multiply $- 1$ by $4$ .

$\frac{6 x^{9} - 4 x^{9}}{{(x^{4})}^{2}}$

Step 3.1.2

Subtract $4 x^{9}$ from $6 x^{9}$ .

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

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, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 x^{9}}{x^{4 \cdot 2}}$

Step 3.2.1.2

Multiply $4$ by $2$ .

$\frac{2 x^{9}}{x^{8}}$

Step 3.2.2

Cancel the common factor of $x^{9}$ and $x^{8}$ .

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Step 3.2.2.1

Factor $x^{8}$ out of $2 x^{9}$ .

$\frac{x^{8} (2 x)}{x^{8}}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

Multiply by $1$ .

$\frac{x^{8} (2 x)}{x^{8} \cdot 1}$

Step 3.2.2.2.2

Cancel the common factor.

$\frac{x^{8} (2 x)}{x^{8} \cdot 1}$

Step 3.2.2.2.3

Rewrite the expression.

$\frac{2 x}{1}$

Step 3.2.2.2.4

Divide $2 x$ by $1$ .

$2 x$