Calculus Examples

$f (x) = \frac{4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5}{x^{3}}$

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 $f (x) = 4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5$ and $g (x) = x^{3}$ .

$\frac{x^{3} \frac{d}{d x} [4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5] - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{{(x^{3})}^{2}}$

Step 2

Differentiate.

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Step 2.1

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 2.1.1

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

$\frac{x^{3} \frac{d}{d x} [4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5] - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 2.1.2

Multiply $3$ by $2$ .

$\frac{x^{3} \frac{d}{d x} [4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5] - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.2

By the Sum Rule, the derivative of $4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5$ with respect to $x$ is $\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]$ .

$\frac{x^{3} (\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.3

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

$\frac{x^{3} (4 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.4

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

$\frac{x^{3} (4 (4 x^{3}) + \frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.5

Multiply $4$ by $4$ .

$\frac{x^{3} (16 x^{3} + \frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.6

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{3}$ with respect to $x$ is $7 \frac{d}{d x} [x^{3}]$ .

$\frac{x^{3} (16 x^{3} + 7 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{x^{3} (16 x^{3} + 7 (3 x^{2}) + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.8

Multiply $3$ by $7$ .

$\frac{x^{3} (16 x^{3} + 21 x^{2} + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.9

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$\frac{x^{3} (16 x^{3} + 21 x^{2} - 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.10

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

$\frac{x^{3} (16 x^{3} + 21 x^{2} - 3 (2 x) + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.11

Multiply $2$ by $- 3$ .

$\frac{x^{3} (16 x^{3} + 21 x^{2} - 6 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.12

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

$\frac{x^{3} (16 x^{3} + 21 x^{2} - 6 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.13

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

$\frac{x^{3} (16 x^{3} + 21 x^{2} - 6 x - 4 \cdot 1 + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.14

Multiply $- 4$ by $1$ .

$\frac{x^{3} (16 x^{3} + 21 x^{2} - 6 x - 4 + \frac{d}{d x} [5]) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.15

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{x^{3} (16 x^{3} + 21 x^{2} - 6 x - 4 + 0) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.16

Add $16 x^{3} + 21 x^{2} - 6 x - 4$ and $0$ .

$\frac{x^{3} (16 x^{3} + 21 x^{2} - 6 x - 4) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.17

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{x^{3} (16 x^{3} + 21 x^{2} - 6 x - 4) - (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) (3 x^{2})}{x^{6}}$

Step 2.18

Simplify with factoring out.

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Step 2.18.1

Multiply $3$ by $- 1$ .

$\frac{x^{3} (16 x^{3} + 21 x^{2} - 6 x - 4) - 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) x^{2}}{x^{6}}$

Step 2.18.2

Factor $x^{2}$ out of $x^{3} (16 x^{3} + 21 x^{2} - 6 x - 4) - 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) x^{2}$ .

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Step 2.18.2.1

Factor $x^{2}$ out of $x^{3} (16 x^{3} + 21 x^{2} - 6 x - 4)$ .

$\frac{x^{2} (x (16 x^{3} + 21 x^{2} - 6 x - 4)) - 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) x^{2}}{x^{6}}$

Step 2.18.2.2

Factor $x^{2}$ out of $- 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5) x^{2}$ .

$\frac{x^{2} (x (16 x^{3} + 21 x^{2} - 6 x - 4)) + x^{2} (- 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5))}{x^{6}}$

Step 2.18.2.3

Factor $x^{2}$ out of $x^{2} (x (16 x^{3} + 21 x^{2} - 6 x - 4)) + x^{2} (- 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5))$ .

$\frac{x^{2} (x (16 x^{3} + 21 x^{2} - 6 x - 4) - 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5))}{x^{6}}$

Step 3

Cancel the common factors.

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Step 3.1

Factor $x^{2}$ out of $x^{6}$ .

$\frac{x^{2} (x (16 x^{3} + 21 x^{2} - 6 x - 4) - 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5))}{x^{2} x^{4}}$

Step 3.2

Cancel the common factor.

$\frac{x^{2} (x (16 x^{3} + 21 x^{2} - 6 x - 4) - 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5))}{x^{2} x^{4}}$

Step 3.3

Rewrite the expression.

$\frac{x (16 x^{3} + 21 x^{2} - 6 x - 4) - 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5)}{x^{4}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{x (16 x^{3}) + x (21 x^{2}) + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4} + 7 x^{3} - 3 x^{2} - 4 x + 5)}{x^{4}}$

Step 4.2

Apply the distributive property.

$\frac{x (16 x^{3}) + x (21 x^{2}) + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{16 x \cdot x^{3} + x (21 x^{2}) + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.2

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

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Step 4.3.1.2.1

Move $x^{3}$ .

$\frac{16 (x^{3} x) + x (21 x^{2}) + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.2.2

Multiply $x^{3}$ by $x$ .

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Step 4.3.1.2.2.1

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

$\frac{16 (x^{3} x^{1}) + x (21 x^{2}) + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.2.2.2

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

$\frac{16 x^{3 + 1} + x (21 x^{2}) + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.2.3

Add $3$ and $1$ .

$\frac{16 x^{4} + x (21 x^{2}) + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{16 x^{4} + 21 x \cdot x^{2} + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.4

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

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Step 4.3.1.4.1

Move $x^{2}$ .

$\frac{16 x^{4} + 21 (x^{2} x) + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.4.2

Multiply $x^{2}$ by $x$ .

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Step 4.3.1.4.2.1

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

$\frac{16 x^{4} + 21 (x^{2} x^{1}) + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.4.2.2

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

$\frac{16 x^{4} + 21 x^{2 + 1} + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.4.3

Add $2$ and $1$ .

$\frac{16 x^{4} + 21 x^{3} + x (- 6 x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.5

Rewrite using the commutative property of multiplication.

$\frac{16 x^{4} + 21 x^{3} - 6 x \cdot x + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.6

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

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Step 4.3.1.6.1

Move $x$ .

$\frac{16 x^{4} + 21 x^{3} - 6 (x \cdot x) + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.6.2

Multiply $x$ by $x$ .

$\frac{16 x^{4} + 21 x^{3} - 6 x^{2} + x \cdot - 4 - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.7

Move $- 4$ to the left of $x$ .

$\frac{16 x^{4} + 21 x^{3} - 6 x^{2} - 4 \cdot x - 3 (4 x^{4}) - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.8

Multiply $4$ by $- 3$ .

$\frac{16 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 12 x^{4} - 3 (7 x^{3}) - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.9

Multiply $7$ by $- 3$ .

$\frac{16 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 12 x^{4} - 21 x^{3} - 3 (- 3 x^{2}) - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.10

Multiply $- 3$ by $- 3$ .

$\frac{16 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 12 x^{4} - 21 x^{3} + 9 x^{2} - 3 (- 4 x) - 3 \cdot 5}{x^{4}}$

Step 4.3.1.11

Multiply $- 4$ by $- 3$ .

$\frac{16 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 12 x^{4} - 21 x^{3} + 9 x^{2} + 12 x - 3 \cdot 5}{x^{4}}$

Step 4.3.1.12

Multiply $- 3$ by $5$ .

$\frac{16 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 12 x^{4} - 21 x^{3} + 9 x^{2} + 12 x - 15}{x^{4}}$

Step 4.3.2

Combine the opposite terms in $16 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 12 x^{4} - 21 x^{3} + 9 x^{2} + 12 x - 15$ .

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Step 4.3.2.1

Subtract $21 x^{3}$ from $21 x^{3}$ .

$\frac{16 x^{4} - 6 x^{2} - 4 x - 12 x^{4} + 0 + 9 x^{2} + 12 x - 15}{x^{4}}$

Step 4.3.2.2

Add $16 x^{4} - 6 x^{2} - 4 x - 12 x^{4}$ and $0$ .

$\frac{16 x^{4} - 6 x^{2} - 4 x - 12 x^{4} + 9 x^{2} + 12 x - 15}{x^{4}}$

Step 4.3.3

Subtract $12 x^{4}$ from $16 x^{4}$ .

$\frac{4 x^{4} - 6 x^{2} - 4 x + 9 x^{2} + 12 x - 15}{x^{4}}$

Step 4.3.4

Add $- 6 x^{2}$ and $9 x^{2}$ .

$\frac{4 x^{4} + 3 x^{2} - 4 x + 12 x - 15}{x^{4}}$

Step 4.3.5

Add $- 4 x$ and $12 x$ .

$\frac{4 x^{4} + 3 x^{2} + 8 x - 15}{x^{4}}$