Calculus Examples

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

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 $f (x) = x^{5} + 1$ and $g (x) = x^{4}$ .

$\frac{x^{4} \frac{d}{d x} [x^{5} + 1] - (x^{5} + 1) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{5} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [1]$ .

$\frac{x^{4} (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [1]) - (x^{5} + 1) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{x^{4} (5 x^{4} + \frac{d}{d x} [1]) - (x^{5} + 1) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 2.3

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

$\frac{x^{4} (5 x^{4} + 0) - (x^{5} + 1) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 2.4

Add $5 x^{4}$ and $0$ .

$\frac{x^{4} (5 x^{4}) - (x^{5} + 1) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 2.5

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x^{4}$ .

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

Step 3.3.1.2.2

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

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

Step 3.3.1.2.3

Add $4$ and $4$ .

$\frac{5 x^{8} - x^{5} (4 x^{3}) - 1 \cdot 1 (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{5 x^{8} - 1 \cdot 4 x^{5} x^{3} - 1 \cdot 1 (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.3.1.4

Multiply $x^{5}$ by $x^{3}$ by adding the exponents.

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Step 3.3.1.4.1

Move $x^{3}$ .

$\frac{5 x^{8} - 1 \cdot 4 (x^{3} x^{5}) - 1 \cdot 1 (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.3.1.4.2

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

$\frac{5 x^{8} - 1 \cdot 4 x^{3 + 5} - 1 \cdot 1 (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.3.1.4.3

Add $3$ and $5$ .

$\frac{5 x^{8} - 1 \cdot 4 x^{8} - 1 \cdot 1 (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.3.1.5

Multiply $- 1$ by $4$ .

$\frac{5 x^{8} - 4 x^{8} - 1 \cdot 1 (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.3.1.6

Multiply $- 1$ by $1$ .

$\frac{5 x^{8} - 4 x^{8} - 1 (4 x^{3})}{{(x^{4})}^{2}}$

Step 3.3.1.7

Multiply $4$ by $- 1$ .

$\frac{5 x^{8} - 4 x^{8} - 4 x^{3}}{{(x^{4})}^{2}}$

Step 3.3.2

Subtract $4 x^{8}$ from $5 x^{8}$ .

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

Step 3.4

Combine terms.

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Step 3.4.1

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

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Step 3.4.1.1

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

$\frac{x^{8} - 4 x^{3}}{x^{4 \cdot 2}}$

Step 3.4.1.2

Multiply $4$ by $2$ .

$\frac{x^{8} - 4 x^{3}}{x^{8}}$

Step 3.4.2

Factor $x^{3}$ out of $x^{8} - 4 x^{3}$ .

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Step 3.4.2.1

Factor $x^{3}$ out of $x^{8}$ .

$\frac{x^{3} x^{5} - 4 x^{3}}{x^{8}}$

Step 3.4.2.2

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

$\frac{x^{3} x^{5} + x^{3} \cdot - 4}{x^{8}}$

Step 3.4.2.3

Factor $x^{3}$ out of $x^{3} x^{5} + x^{3} \cdot - 4$ .

$\frac{x^{3} (x^{5} - 4)}{x^{8}}$

Step 3.4.3

Cancel the common factors.

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Step 3.4.3.1

Factor $x^{3}$ out of $x^{8}$ .

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

Step 3.4.3.2

Cancel the common factor.

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

Step 3.4.3.3

Rewrite the expression.

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