Calculus Examples

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

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 $f (z) = 4 z^{4} + 2 z^{2} + 1$ and $g (z) = 2 z^{3} - z$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $3$ by $2$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply $- 1$ by $1$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Multiply $4$ by $4$ .

$(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (2 z^{3} - z) (16 z^{3} + \frac{d}{d z} [2 z^{2}] + \frac{d}{d z} [1])$

Step 2.12

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

$(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (2 z^{3} - z) (16 z^{3} + 2 \frac{d}{d z} [z^{2}] + \frac{d}{d z} [1])$

Step 2.13

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

$(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (2 z^{3} - z) (16 z^{3} + 2 (2 z) + \frac{d}{d z} [1])$

Step 2.14

Multiply $2$ by $2$ .

$(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (2 z^{3} - z) (16 z^{3} + 4 z + \frac{d}{d z} [1])$

Step 2.15

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

$(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (2 z^{3} - z) (16 z^{3} + 4 z + 0)$

Step 2.16

Add $16 z^{3} + 4 z$ and $0$ .

$(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (2 z^{3} - z) (16 z^{3} + 4 z)$

Step 3

Simplify.

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Step 3.1

Factor $1$ out of $(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (2 z^{3} - z) (16 z^{3} + 4 z)$ .

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Step 3.1.1

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

$1 ((4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1)) + (2 z^{3} - z) (16 z^{3} + 4 z)$

Step 3.1.2

Factor $1$ out of $(2 z^{3} - z) (16 z^{3} + 4 z)$ .

$1 ((4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1)) + 1 ((2 z^{3} - z) (16 z^{3} + 4 z))$

Step 3.1.3

Factor $1$ out of $1 ((4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1)) + 1 ((2 z^{3} - z) (16 z^{3} + 4 z))$ .

$1 ((4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (2 z^{3} - z) (16 z^{3} + 4 z))$

Step 3.2

Multiply $(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (2 z^{3} - z) (16 z^{3} + 4 z)$ by $1$ .

$(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (2 z^{3} - z) (16 z^{3} + 4 z)$

Step 3.3

Reorder terms.

$(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1) + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4

Simplify each term.

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Step 3.4.1

Expand $(4 z^{4} + 2 z^{2} + 1) (6 z^{2} - 1)$ by multiplying each term in the first expression by each term in the second expression.

$4 z^{4} (6 z^{2}) + 4 z^{4} \cdot - 1 + 2 z^{2} (6 z^{2}) + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2

Simplify each term.

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Step 3.4.2.1

Rewrite using the commutative property of multiplication.

$4 \cdot 6 z^{4} z^{2} + 4 z^{4} \cdot - 1 + 2 z^{2} (6 z^{2}) + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.2

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

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Step 3.4.2.2.1

Move $z^{2}$ .

$4 \cdot 6 (z^{2} z^{4}) + 4 z^{4} \cdot - 1 + 2 z^{2} (6 z^{2}) + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.2.2

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

$4 \cdot 6 z^{2 + 4} + 4 z^{4} \cdot - 1 + 2 z^{2} (6 z^{2}) + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.2.3

Add $2$ and $4$ .

$4 \cdot 6 z^{6} + 4 z^{4} \cdot - 1 + 2 z^{2} (6 z^{2}) + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.3

Multiply $4$ by $6$ .

$24 z^{6} + 4 z^{4} \cdot - 1 + 2 z^{2} (6 z^{2}) + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.4

Multiply $- 1$ by $4$ .

$24 z^{6} - 4 z^{4} + 2 z^{2} (6 z^{2}) + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.5

Rewrite using the commutative property of multiplication.

$24 z^{6} - 4 z^{4} + 2 \cdot 6 z^{2} z^{2} + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.6

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

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Step 3.4.2.6.1

Move $z^{2}$ .

$24 z^{6} - 4 z^{4} + 2 \cdot 6 (z^{2} z^{2}) + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.6.2

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

$24 z^{6} - 4 z^{4} + 2 \cdot 6 z^{2 + 2} + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.6.3

Add $2$ and $2$ .

$24 z^{6} - 4 z^{4} + 2 \cdot 6 z^{4} + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.7

Multiply $2$ by $6$ .

$24 z^{6} - 4 z^{4} + 12 z^{4} + 2 z^{2} \cdot - 1 + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.8

Multiply $- 1$ by $2$ .

$24 z^{6} - 4 z^{4} + 12 z^{4} - 2 z^{2} + 1 (6 z^{2}) + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.9

Multiply $6 z^{2}$ by $1$ .

$24 z^{6} - 4 z^{4} + 12 z^{4} - 2 z^{2} + 6 z^{2} + 1 \cdot - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.2.10

Multiply $- 1$ by $1$ .

$24 z^{6} - 4 z^{4} + 12 z^{4} - 2 z^{2} + 6 z^{2} - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.3

Add $- 4 z^{4}$ and $12 z^{4}$ .

$24 z^{6} + 8 z^{4} - 2 z^{2} + 6 z^{2} - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.4

Add $- 2 z^{2}$ and $6 z^{2}$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + (16 z^{3} + 4 z) (2 z^{3} - z)$

Step 3.4.5

Expand $(16 z^{3} + 4 z) (2 z^{3} - z)$ using the FOIL Method.

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Step 3.4.5.1

Apply the distributive property.

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 16 z^{3} (2 z^{3} - z) + 4 z (2 z^{3} - z)$

Step 3.4.5.2

Apply the distributive property.

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 16 z^{3} (2 z^{3}) + 16 z^{3} (- z) + 4 z (2 z^{3} - z)$

Step 3.4.5.3

Apply the distributive property.

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 16 z^{3} (2 z^{3}) + 16 z^{3} (- z) + 4 z (2 z^{3}) + 4 z (- z)$

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.

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 16 \cdot 2 z^{3} z^{3} + 16 z^{3} (- z) + 4 z (2 z^{3}) + 4 z (- z)$

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 $z^{3}$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 16 \cdot 2 (z^{3} z^{3}) + 16 z^{3} (- z) + 4 z (2 z^{3}) + 4 z (- z)$

Step 3.4.6.1.2.2

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

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 16 \cdot 2 z^{3 + 3} + 16 z^{3} (- z) + 4 z (2 z^{3}) + 4 z (- z)$

Step 3.4.6.1.2.3

Add $3$ and $3$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 16 \cdot 2 z^{6} + 16 z^{3} (- z) + 4 z (2 z^{3}) + 4 z (- z)$

Step 3.4.6.1.3

Multiply $16$ by $2$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} + 16 z^{3} (- z) + 4 z (2 z^{3}) + 4 z (- z)$

Step 3.4.6.1.4

Rewrite using the commutative property of multiplication.

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} + 16 \cdot - 1 z^{3} z + 4 z (2 z^{3}) + 4 z (- z)$

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 $z$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} + 16 \cdot - 1 (z \cdot z^{3}) + 4 z (2 z^{3}) + 4 z (- z)$

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 $1$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} + 16 \cdot - 1 (z^{1} z^{3}) + 4 z (2 z^{3}) + 4 z (- z)$

Step 3.4.6.1.5.2.2

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

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} + 16 \cdot - 1 z^{1 + 3} + 4 z (2 z^{3}) + 4 z (- z)$

Step 3.4.6.1.5.3

Add $1$ and $3$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} + 16 \cdot - 1 z^{4} + 4 z (2 z^{3}) + 4 z (- z)$

Step 3.4.6.1.6

Multiply $16$ by $- 1$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 4 z (2 z^{3}) + 4 z (- z)$

Step 3.4.6.1.7

Rewrite using the commutative property of multiplication.

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 4 \cdot 2 z \cdot z^{3} + 4 z (- z)$

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 $z^{3}$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 4 \cdot 2 (z^{3} z) + 4 z (- z)$

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 $1$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 4 \cdot 2 (z^{3} z^{1}) + 4 z (- z)$

Step 3.4.6.1.8.2.2

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

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 4 \cdot 2 z^{3 + 1} + 4 z (- z)$

Step 3.4.6.1.8.3

Add $3$ and $1$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 4 \cdot 2 z^{4} + 4 z (- z)$

Step 3.4.6.1.9

Multiply $4$ by $2$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 8 z^{4} + 4 z (- z)$

Step 3.4.6.1.10

Rewrite using the commutative property of multiplication.

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 8 z^{4} + 4 \cdot - 1 z \cdot z$

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 $z$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 8 z^{4} + 4 \cdot - 1 (z \cdot z)$

Step 3.4.6.1.11.2

Multiply $z$ by $z$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 8 z^{4} + 4 \cdot - 1 z^{2}$

Step 3.4.6.1.12

Multiply $4$ by $- 1$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 16 z^{4} + 8 z^{4} - 4 z^{2}$

Step 3.4.6.2

Add $- 16 z^{4}$ and $8 z^{4}$ .

$24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 8 z^{4} - 4 z^{2}$

Step 3.5

Combine the opposite terms in $24 z^{6} + 8 z^{4} + 4 z^{2} - 1 + 32 z^{6} - 8 z^{4} - 4 z^{2}$ .

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Step 3.5.1

Subtract $8 z^{4}$ from $8 z^{4}$ .

$24 z^{6} + 4 z^{2} - 1 + 32 z^{6} + 0 - 4 z^{2}$

Step 3.5.2

Add $24 z^{6} + 4 z^{2} - 1 + 32 z^{6}$ and $0$ .

$24 z^{6} + 4 z^{2} - 1 + 32 z^{6} - 4 z^{2}$

Step 3.5.3

Subtract $4 z^{2}$ from $4 z^{2}$ .

$24 z^{6} - 1 + 32 z^{6} + 0$

Step 3.5.4

Add $24 z^{6} - 1 + 32 z^{6}$ and $0$ .

$24 z^{6} - 1 + 32 z^{6}$

Step 3.6

Add $24 z^{6}$ and $32 z^{6}$ .

$56 z^{6} - 1$