Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{- 4 x^{6} + 4 x^{5} - 3 x^{3}}{3 x^{2}}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\frac{- 4 x^{6} + 4 x^{5} - 3 x^{3}}{x^{2}}]$ .

$\frac{1}{3} \frac{d}{d x} [\frac{- 4 x^{6} + 4 x^{5} - 3 x^{3}}{x^{2}}]$

Step 1.2

Simplify terms.

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Step 1.2.1

Factor $x^{3}$ out of $- 4 x^{6} + 4 x^{5} - 3 x^{3}$ .

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Step 1.2.1.1

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

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

Step 1.2.1.2

Factor $x^{3}$ out of $4 x^{5}$ .

$\frac{1}{3} \frac{d}{d x} [\frac{x^{3} (- 4 x^{3}) + x^{3} (4 x^{2}) - 3 x^{3}}{x^{2}}]$

Step 1.2.1.3

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

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

Step 1.2.1.4

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

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

Step 1.2.1.5

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

$\frac{1}{3} \frac{d}{d x} [\frac{x^{3} (- 4 x^{3} + 4 x^{2} - 3)}{x^{2}}]$

Step 1.2.2

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 1.2.2.1

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

$\frac{1}{3} \frac{d}{d x} [\frac{x^{2} (x (- 4 x^{3} + 4 x^{2} - 3))}{x^{2}}]$

Step 1.2.2.2

Cancel the common factors.

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Step 1.2.2.2.1

Multiply by $1$ .

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

Step 1.2.2.2.2

Cancel the common factor.

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

Step 1.2.2.2.3

Rewrite the expression.

$\frac{1}{3} \frac{d}{d x} [\frac{x (- 4 x^{3} + 4 x^{2} - 3)}{1}]$

Step 1.2.2.2.4

Divide $x (- 4 x^{3} + 4 x^{2} - 3)$ by $1$ .

$\frac{1}{3} \frac{d}{d x} [x (- 4 x^{3} + 4 x^{2} - 3)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = - 4 x^{3} + 4 x^{2} - 3$ .

$\frac{1}{3} (x \frac{d}{d x} [- 4 x^{3} + 4 x^{2} - 3] + (- 4 x^{3} + 4 x^{2} - 3) \frac{d}{d x} [x])$

Step 3

Differentiate.

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Step 3.1

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

$\frac{1}{3} (x (\frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 3]) + (- 4 x^{3} + 4 x^{2} - 3) \frac{d}{d x} [x])$

Step 3.2

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

$\frac{1}{3} (x (- 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 3]) + (- 4 x^{3} + 4 x^{2} - 3) \frac{d}{d x} [x])$

Step 3.3

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

$\frac{1}{3} (x (- 4 (3 x^{2}) + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 3]) + (- 4 x^{3} + 4 x^{2} - 3) \frac{d}{d x} [x])$

Step 3.4

Multiply $3$ by $- 4$ .

$\frac{1}{3} (x (- 12 x^{2} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 3]) + (- 4 x^{3} + 4 x^{2} - 3) \frac{d}{d x} [x])$

Step 3.5

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{3} (x (- 12 x^{2} + 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]) + (- 4 x^{3} + 4 x^{2} - 3) \frac{d}{d x} [x])$

Step 3.6

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

$\frac{1}{3} (x (- 12 x^{2} + 4 (2 x) + \frac{d}{d x} [- 3]) + (- 4 x^{3} + 4 x^{2} - 3) \frac{d}{d x} [x])$

Step 3.7

Multiply $2$ by $4$ .

$\frac{1}{3} (x (- 12 x^{2} + 8 x + \frac{d}{d x} [- 3]) + (- 4 x^{3} + 4 x^{2} - 3) \frac{d}{d x} [x])$

Step 3.8

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{1}{3} (x (- 12 x^{2} + 8 x + 0) + (- 4 x^{3} + 4 x^{2} - 3) \frac{d}{d x} [x])$

Step 3.9

Add $- 12 x^{2} + 8 x$ and $0$ .

$\frac{1}{3} (x (- 12 x^{2} + 8 x) + (- 4 x^{3} + 4 x^{2} - 3) \frac{d}{d x} [x])$

Step 3.10

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

$\frac{1}{3} (x (- 12 x^{2} + 8 x) + (- 4 x^{3} + 4 x^{2} - 3) \cdot 1)$

Step 3.11

Multiply $- 4 x^{3} + 4 x^{2} - 3$ by $1$ .

$\frac{1}{3} (x (- 12 x^{2} + 8 x) - 4 x^{3} + 4 x^{2} - 3)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{1}{3} (x (- 12 x^{2}) + x (8 x) - 4 x^{3} + 4 x^{2} - 3)$

Step 4.2

Apply the distributive property.

$\frac{1}{3} (x (- 12 x^{2})) + \frac{1}{3} (x (8 x)) + \frac{1}{3} (- 4 x^{3}) + \frac{1}{3} (4 x^{2}) + \frac{1}{3} \cdot - 3$

Step 4.3

Combine terms.

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Step 4.3.1

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

$\frac{1}{3} (- 12 (x^{1} x^{2})) + \frac{1}{3} (x (8 x)) + \frac{1}{3} (- 4 x^{3}) + \frac{1}{3} (4 x^{2}) + \frac{1}{3} \cdot - 3$

Step 4.3.2

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

$\frac{1}{3} (- 12 x^{1 + 2}) + \frac{1}{3} (x (8 x)) + \frac{1}{3} (- 4 x^{3}) + \frac{1}{3} (4 x^{2}) + \frac{1}{3} \cdot - 3$

Step 4.3.3

Add $1$ and $2$ .

$\frac{1}{3} (- 12 x^{3}) + \frac{1}{3} (x (8 x)) + \frac{1}{3} (- 4 x^{3}) + \frac{1}{3} (4 x^{2}) + \frac{1}{3} \cdot - 3$

Step 4.3.4

Combine $- 12$ and $\frac{1}{3}$ .

$\frac{- 12}{3} x^{3} + \frac{1}{3} (x (8 x)) + \frac{1}{3} (- 4 x^{3}) + \frac{1}{3} (4 x^{2}) + \frac{1}{3} \cdot - 3$

Step 4.3.5

Combine $\frac{- 12}{3}$ and $x^{3}$ .

$\frac{- 12 x^{3}}{3} + \frac{1}{3} (x (8 x)) + \frac{1}{3} (- 4 x^{3}) + \frac{1}{3} (4 x^{2}) + \frac{1}{3} \cdot - 3$

Step 4.3.6

Cancel the common factor of $- 12$ and $3$ .

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Step 4.3.6.1

Factor $3$ out of $- 12 x^{3}$ .

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

Step 4.3.6.2

Cancel the common factors.

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Step 4.3.6.2.1

Factor $3$ out of $3$ .

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

Step 4.3.6.2.2

Cancel the common factor.

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

Step 4.3.6.2.3

Rewrite the expression.

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

Step 4.3.6.2.4

Divide $- 4 x^{3}$ by $1$ .

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

Step 4.3.7

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

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

Step 4.3.8

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

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

Step 4.3.9

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

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

Step 4.3.10

Add $1$ and $1$ .

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

Step 4.3.11

Combine $8$ and $\frac{1}{3}$ .

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

Step 4.3.12

Combine $\frac{8}{3}$ and $x^{2}$ .

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

Step 4.3.13

Combine $- 4$ and $\frac{1}{3}$ .

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

Step 4.3.14

Combine $\frac{- 4}{3}$ and $x^{3}$ .

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

Step 4.3.15

Move the negative in front of the fraction.

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

Step 4.3.16

To write $- 4 x^{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.3.17

Combine $- 4 x^{3}$ and $\frac{3}{3}$ .

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

Step 4.3.18

Combine the numerators over the common denominator.

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

Step 4.3.19

Multiply $3$ by $- 4$ .

$\frac{8 x^{2}}{3} + \frac{- 12 x^{3} - 4 x^{3}}{3} + \frac{1}{3} (4 x^{2}) + \frac{1}{3} \cdot - 3$

Step 4.3.20

Subtract $4 x^{3}$ from $- 12 x^{3}$ .

$\frac{8 x^{2}}{3} + \frac{- 16 x^{3}}{3} + \frac{1}{3} (4 x^{2}) + \frac{1}{3} \cdot - 3$

Step 4.3.21

Move the negative in front of the fraction.

$\frac{8 x^{2}}{3} - \frac{16 x^{3}}{3} + \frac{1}{3} (4 x^{2}) + \frac{1}{3} \cdot - 3$

Step 4.3.22

Combine $4$ and $\frac{1}{3}$ .

$\frac{8 x^{2}}{3} - \frac{16 x^{3}}{3} + \frac{4}{3} x^{2} + \frac{1}{3} \cdot - 3$

Step 4.3.23

Combine $\frac{4}{3}$ and $x^{2}$ .

$\frac{8 x^{2}}{3} - \frac{16 x^{3}}{3} + \frac{4 x^{2}}{3} + \frac{1}{3} \cdot - 3$

Step 4.3.24

Combine the numerators over the common denominator.

$- \frac{16 x^{3}}{3} + \frac{8 x^{2} + 4 x^{2}}{3} + \frac{1}{3} \cdot - 3$

Step 4.3.25

Add $8 x^{2}$ and $4 x^{2}$ .

$- \frac{16 x^{3}}{3} + \frac{12 x^{2}}{3} + \frac{1}{3} \cdot - 3$

Step 4.3.26

Cancel the common factor of $12$ and $3$ .

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Step 4.3.26.1

Factor $3$ out of $12 x^{2}$ .

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

Step 4.3.26.2

Cancel the common factors.

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Step 4.3.26.2.1

Factor $3$ out of $3$ .

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

Step 4.3.26.2.2

Cancel the common factor.

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

Step 4.3.26.2.3

Rewrite the expression.

$- \frac{16 x^{3}}{3} + \frac{4 x^{2}}{1} + \frac{1}{3} \cdot - 3$

Step 4.3.26.2.4

Divide $4 x^{2}$ by $1$ .

$- \frac{16 x^{3}}{3} + 4 x^{2} + \frac{1}{3} \cdot - 3$

Step 4.3.27

Combine $\frac{1}{3}$ and $- 3$ .

$- \frac{16 x^{3}}{3} + 4 x^{2} + \frac{- 3}{3}$

Step 4.3.28

Cancel the common factor of $- 3$ and $3$ .

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Step 4.3.28.1

Factor $3$ out of $- 3$ .

$- \frac{16 x^{3}}{3} + 4 x^{2} + \frac{3 \cdot - 1}{3}$

Step 4.3.28.2

Cancel the common factors.

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Step 4.3.28.2.1

Factor $3$ out of $3$ .

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

Step 4.3.28.2.2

Cancel the common factor.

$- \frac{16 x^{3}}{3} + 4 x^{2} + \frac{3 \cdot - 1}{3 \cdot 1}$

Step 4.3.28.2.3

Rewrite the expression.

$- \frac{16 x^{3}}{3} + 4 x^{2} + \frac{- 1}{1}$

Step 4.3.28.2.4

Divide $- 1$ by $1$ .

$- \frac{16 x^{3}}{3} + 4 x^{2} - 1$