Calculus Examples

$f (z) = (9 z^{4} + 3 z^{2} + 1) (3 z^{3} - z)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d z} [f (z) g (z)]$ is $f (z) \frac{d}{d z} [g (z)] + g (z) \frac{d}{d z} [f (z)]$ where $f (z) = 9 z^{4} + 3 z^{2} + 1$ and $g (z) = 3 z^{3} - z$ .

$(9 z^{4} + 3 z^{2} + 1) \frac{d}{d z} [3 z^{3} - z] + (3 z^{3} - z) \frac{d}{d z} [9 z^{4} + 3 z^{2} + 1]$

Step 2

Differentiate.

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Step 2.1

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

$(9 z^{4} + 3 z^{2} + 1) (\frac{d}{d z} [3 z^{3}] + \frac{d}{d z} [- z]) + (3 z^{3} - z) \frac{d}{d z} [9 z^{4} + 3 z^{2} + 1]$

Step 2.2

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

$(9 z^{4} + 3 z^{2} + 1) (3 \frac{d}{d z} [z^{3}] + \frac{d}{d z} [- z]) + (3 z^{3} - z) \frac{d}{d z} [9 z^{4} + 3 z^{2} + 1]$

Step 2.3

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

$(9 z^{4} + 3 z^{2} + 1) (3 (3 z^{2}) + \frac{d}{d z} [- z]) + (3 z^{3} - z) \frac{d}{d z} [9 z^{4} + 3 z^{2} + 1]$

Step 2.4

Multiply $3$ by $3$ .

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} + \frac{d}{d z} [- z]) + (3 z^{3} - z) \frac{d}{d z} [9 z^{4} + 3 z^{2} + 1]$

Step 2.5

Since $- 1$ is constant with respect to $z$ , the derivative of $- z$ with respect to $z$ is $- \frac{d}{d z} [z]$ .

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - \frac{d}{d z} [z]) + (3 z^{3} - z) \frac{d}{d z} [9 z^{4} + 3 z^{2} + 1]$

Step 2.6

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

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1 \cdot 1) + (3 z^{3} - z) \frac{d}{d z} [9 z^{4} + 3 z^{2} + 1]$

Step 2.7

Multiply $- 1$ by $1$ .

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) \frac{d}{d z} [9 z^{4} + 3 z^{2} + 1]$

Step 2.8

By the Sum Rule, the derivative of $9 z^{4} + 3 z^{2} + 1$ with respect to $z$ is $\frac{d}{d z} [9 z^{4}] + \frac{d}{d z} [3 z^{2}] + \frac{d}{d z} [1]$ .

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (\frac{d}{d z} [9 z^{4}] + \frac{d}{d z} [3 z^{2}] + \frac{d}{d z} [1])$

Step 2.9

Since $9$ is constant with respect to $z$ , the derivative of $9 z^{4}$ with respect to $z$ is $9 \frac{d}{d z} [z^{4}]$ .

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (9 \frac{d}{d z} [z^{4}] + \frac{d}{d z} [3 z^{2}] + \frac{d}{d z} [1])$

Step 2.10

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

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (9 (4 z^{3}) + \frac{d}{d z} [3 z^{2}] + \frac{d}{d z} [1])$

Step 2.11

Multiply $4$ by $9$ .

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (36 z^{3} + \frac{d}{d z} [3 z^{2}] + \frac{d}{d z} [1])$

Step 2.12

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

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (36 z^{3} + 3 \frac{d}{d z} [z^{2}] + \frac{d}{d z} [1])$

Step 2.13

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

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (36 z^{3} + 3 (2 z) + \frac{d}{d z} [1])$

Step 2.14

Multiply $2$ by $3$ .

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (36 z^{3} + 6 z + \frac{d}{d z} [1])$

Step 2.15

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

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (36 z^{3} + 6 z + 0)$

Step 2.16

Add $36 z^{3} + 6 z$ and $0$ .

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (36 z^{3} + 6 z)$

Step 3

Simplify.

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Step 3.1

Factor $1$ out of $(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (36 z^{3} + 6 z)$ .

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Step 3.1.1

Factor $1$ out of $(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1)$ .

$1 ((9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1)) + (3 z^{3} - z) (36 z^{3} + 6 z)$

Step 3.1.2

Factor $1$ out of $(3 z^{3} - z) (36 z^{3} + 6 z)$ .

$1 ((9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1)) + 1 ((3 z^{3} - z) (36 z^{3} + 6 z))$

Step 3.1.3

Factor $1$ out of $1 ((9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1)) + 1 ((3 z^{3} - z) (36 z^{3} + 6 z))$ .

$1 ((9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (36 z^{3} + 6 z))$

Step 3.2

Multiply $(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (36 z^{3} + 6 z)$ by $1$ .

$(9 z^{4} + 3 z^{2} + 1) (9 z^{2} - 1) + (3 z^{3} - z) (36 z^{3} + 6 z)$

Step 3.3

Reorder terms.

$(9 z^{2} - 1) (9 z^{4} + 3 z^{2} + 1) + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4

Simplify each term.

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Step 3.4.1

Expand $(9 z^{2} - 1) (9 z^{4} + 3 z^{2} + 1)$ by multiplying each term in the first expression by each term in the second expression.

$9 z^{2} (9 z^{4}) + 9 z^{2} (3 z^{2}) + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2

Simplify each term.

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Step 3.4.2.1

Rewrite using the commutative property of multiplication.

$9 \cdot 9 z^{2} z^{4} + 9 z^{2} (3 z^{2}) + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.2

Multiply $z^{2}$ by $z^{4}$ by adding the exponents.

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Step 3.4.2.2.1

Move $z^{4}$ .

$9 \cdot 9 (z^{4} z^{2}) + 9 z^{2} (3 z^{2}) + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.2.2

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

$9 \cdot 9 z^{4 + 2} + 9 z^{2} (3 z^{2}) + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.2.3

Add $4$ and $2$ .

$9 \cdot 9 z^{6} + 9 z^{2} (3 z^{2}) + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.3

Multiply $9$ by $9$ .

$81 z^{6} + 9 z^{2} (3 z^{2}) + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.4

Rewrite using the commutative property of multiplication.

$81 z^{6} + 9 \cdot 3 z^{2} z^{2} + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.5

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

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Step 3.4.2.5.1

Move $z^{2}$ .

$81 z^{6} + 9 \cdot 3 (z^{2} z^{2}) + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.5.2

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

$81 z^{6} + 9 \cdot 3 z^{2 + 2} + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.5.3

Add $2$ and $2$ .

$81 z^{6} + 9 \cdot 3 z^{4} + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.6

Multiply $9$ by $3$ .

$81 z^{6} + 27 z^{4} + 9 z^{2} \cdot 1 - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.7

Multiply $9$ by $1$ .

$81 z^{6} + 27 z^{4} + 9 z^{2} - 1 (9 z^{4}) - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.8

Multiply $9$ by $- 1$ .

$81 z^{6} + 27 z^{4} + 9 z^{2} - 9 z^{4} - 1 (3 z^{2}) - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.9

Multiply $3$ by $- 1$ .

$81 z^{6} + 27 z^{4} + 9 z^{2} - 9 z^{4} - 3 z^{2} - 1 \cdot 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.2.10

Multiply $- 1$ by $1$ .

$81 z^{6} + 27 z^{4} + 9 z^{2} - 9 z^{4} - 3 z^{2} - 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.3

Subtract $9 z^{4}$ from $27 z^{4}$ .

$81 z^{6} + 18 z^{4} + 9 z^{2} - 3 z^{2} - 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.4

Subtract $3 z^{2}$ from $9 z^{2}$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + (36 z^{3} + 6 z) (3 z^{3} - z)$

Step 3.4.5

Expand $(36 z^{3} + 6 z) (3 z^{3} - z)$ using the FOIL Method.

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Step 3.4.5.1

Apply the distributive property.

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 36 z^{3} (3 z^{3} - z) + 6 z (3 z^{3} - z)$

Step 3.4.5.2

Apply the distributive property.

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 36 z^{3} (3 z^{3}) + 36 z^{3} (- z) + 6 z (3 z^{3} - z)$

Step 3.4.5.3

Apply the distributive property.

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 36 z^{3} (3 z^{3}) + 36 z^{3} (- z) + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6

Simplify and combine like terms.

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Step 3.4.6.1

Simplify each term.

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Step 3.4.6.1.1

Rewrite using the commutative property of multiplication.

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 36 \cdot 3 z^{3} z^{3} + 36 z^{3} (- z) + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.2

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

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Step 3.4.6.1.2.1

Move $z^{3}$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 36 \cdot 3 (z^{3} z^{3}) + 36 z^{3} (- z) + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.2.2

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

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 36 \cdot 3 z^{3 + 3} + 36 z^{3} (- z) + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.2.3

Add $3$ and $3$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 36 \cdot 3 z^{6} + 36 z^{3} (- z) + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.3

Multiply $36$ by $3$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} + 36 z^{3} (- z) + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.4

Rewrite using the commutative property of multiplication.

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} + 36 \cdot - 1 z^{3} z + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.5

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

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Step 3.4.6.1.5.1

Move $z$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} + 36 \cdot - 1 (z \cdot z^{3}) + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.5.2

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

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Step 3.4.6.1.5.2.1

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

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} + 36 \cdot - 1 (z^{1} z^{3}) + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.5.2.2

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

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} + 36 \cdot - 1 z^{1 + 3} + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.5.3

Add $1$ and $3$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} + 36 \cdot - 1 z^{4} + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.6

Multiply $36$ by $- 1$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 6 z (3 z^{3}) + 6 z (- z)$

Step 3.4.6.1.7

Rewrite using the commutative property of multiplication.

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 6 \cdot 3 z \cdot z^{3} + 6 z (- z)$

Step 3.4.6.1.8

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

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Step 3.4.6.1.8.1

Move $z^{3}$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 6 \cdot 3 (z^{3} z) + 6 z (- z)$

Step 3.4.6.1.8.2

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

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Step 3.4.6.1.8.2.1

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

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 6 \cdot 3 (z^{3} z^{1}) + 6 z (- z)$

Step 3.4.6.1.8.2.2

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

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 6 \cdot 3 z^{3 + 1} + 6 z (- z)$

Step 3.4.6.1.8.3

Add $3$ and $1$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 6 \cdot 3 z^{4} + 6 z (- z)$

Step 3.4.6.1.9

Multiply $6$ by $3$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 18 z^{4} + 6 z (- z)$

Step 3.4.6.1.10

Rewrite using the commutative property of multiplication.

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 18 z^{4} + 6 \cdot - 1 z \cdot z$

Step 3.4.6.1.11

Multiply $z$ by $z$ by adding the exponents.

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Step 3.4.6.1.11.1

Move $z$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 18 z^{4} + 6 \cdot - 1 (z \cdot z)$

Step 3.4.6.1.11.2

Multiply $z$ by $z$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 18 z^{4} + 6 \cdot - 1 z^{2}$

Step 3.4.6.1.12

Multiply $6$ by $- 1$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 36 z^{4} + 18 z^{4} - 6 z^{2}$

Step 3.4.6.2

Add $- 36 z^{4}$ and $18 z^{4}$ .

$81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 18 z^{4} - 6 z^{2}$

Step 3.5

Combine the opposite terms in $81 z^{6} + 18 z^{4} + 6 z^{2} - 1 + 108 z^{6} - 18 z^{4} - 6 z^{2}$ .

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Step 3.5.1

Subtract $18 z^{4}$ from $18 z^{4}$ .

$81 z^{6} + 6 z^{2} - 1 + 108 z^{6} + 0 - 6 z^{2}$

Step 3.5.2

Add $81 z^{6} + 6 z^{2} - 1 + 108 z^{6}$ and $0$ .

$81 z^{6} + 6 z^{2} - 1 + 108 z^{6} - 6 z^{2}$

Step 3.5.3

Subtract $6 z^{2}$ from $6 z^{2}$ .

$81 z^{6} - 1 + 108 z^{6} + 0$

Step 3.5.4

Add $81 z^{6} - 1 + 108 z^{6}$ and $0$ .

$81 z^{6} - 1 + 108 z^{6}$

Step 3.6

Add $81 z^{6}$ and $108 z^{6}$ .

$189 z^{6} - 1$