Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $- 2$ by $1$ .

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

Step 2.6

Since $2$ is constant with respect to $z$ , the derivative of $2$ with respect to $z$ is $0$ .

$(4 - z^{2}) (3 z^{2} - 2 + 0) + (z^{3} - 2 z + 2) \frac{d}{d z} [4 - z^{2}]$

Step 2.7

Add $3 z^{2} - 2$ and $0$ .

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

Step 2.8

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

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

Step 2.9

Since $4$ is constant with respect to $z$ , the derivative of $4$ with respect to $z$ is $0$ .

$(4 - z^{2}) (3 z^{2} - 2) + (z^{3} - 2 z + 2) (0 + \frac{d}{d z} [- z^{2}])$

Step 2.10

Add $0$ and $\frac{d}{d z} [- z^{2}]$ .

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

Step 2.11

Since $- 1$ is constant with respect to $z$ , the derivative of $- z^{2}$ with respect to $z$ is $- \frac{d}{d z} [z^{2}]$ .

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

Step 2.12

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

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

Step 2.13

Multiply $2$ by $- 1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Combine terms.

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Step 3.5.1

Multiply $3$ by $4$ .

$12 z^{2} - z^{2} (3 z^{2}) + 4 \cdot - 2 - z^{2} \cdot - 2 + z^{3} (- 2 z) - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.2

Multiply $3$ by $- 1$ .

$12 z^{2} - 3 z^{2} z^{2} + 4 \cdot - 2 - z^{2} \cdot - 2 + z^{3} (- 2 z) - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.3

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

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Step 3.5.3.1

Move $z^{2}$ .

$12 z^{2} - 3 (z^{2} z^{2}) + 4 \cdot - 2 - z^{2} \cdot - 2 + z^{3} (- 2 z) - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.3.2

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

$12 z^{2} - 3 z^{2 + 2} + 4 \cdot - 2 - z^{2} \cdot - 2 + z^{3} (- 2 z) - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.3.3

Add $2$ and $2$ .

$12 z^{2} - 3 z^{4} + 4 \cdot - 2 - z^{2} \cdot - 2 + z^{3} (- 2 z) - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.4

Multiply $4$ by $- 2$ .

$12 z^{2} - 3 z^{4} - 8 - z^{2} \cdot - 2 + z^{3} (- 2 z) - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.5

Multiply $- 2$ by $- 1$ .

$12 z^{2} - 3 z^{4} - 8 + 2 z^{2} + z^{3} (- 2 z) - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.6

Add $12 z^{2}$ and $2 z^{2}$ .

$14 z^{2} - 3 z^{4} - 8 + z^{3} (- 2 z) - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.7

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

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Step 3.5.7.1

Move $z$ .

$14 z^{2} - 3 z^{4} - 8 + z \cdot z^{3} \cdot - 2 - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.7.2

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

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Step 3.5.7.2.1

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

$14 z^{2} - 3 z^{4} - 8 + z^{1} z^{3} \cdot - 2 - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.7.2.2

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

$14 z^{2} - 3 z^{4} - 8 + z^{1 + 3} \cdot - 2 - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.7.3

Add $1$ and $3$ .

$14 z^{2} - 3 z^{4} - 8 + z^{4} \cdot - 2 - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.8

Move $- 2$ to the left of $z^{4}$ .

$14 z^{2} - 3 z^{4} - 8 - 2 \cdot z^{4} - 2 z (- 2 z) + 2 (- 2 z)$

Step 3.5.9

Multiply $- 2$ by $- 2$ .

$14 z^{2} - 3 z^{4} - 8 - 2 z^{4} + 4 z \cdot z + 2 (- 2 z)$

Step 3.5.10

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

$14 z^{2} - 3 z^{4} - 8 - 2 z^{4} + 4 (z^{1} z) + 2 (- 2 z)$

Step 3.5.11

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

$14 z^{2} - 3 z^{4} - 8 - 2 z^{4} + 4 (z^{1} z^{1}) + 2 (- 2 z)$

Step 3.5.12

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

$14 z^{2} - 3 z^{4} - 8 - 2 z^{4} + 4 z^{1 + 1} + 2 (- 2 z)$

Step 3.5.13

Add $1$ and $1$ .

$14 z^{2} - 3 z^{4} - 8 - 2 z^{4} + 4 z^{2} + 2 (- 2 z)$

Step 3.5.14

Multiply $- 2$ by $2$ .

$14 z^{2} - 3 z^{4} - 8 - 2 z^{4} + 4 z^{2} - 4 z$

Step 3.5.15

Subtract $2 z^{4}$ from $- 3 z^{4}$ .

$14 z^{2} - 5 z^{4} - 8 + 4 z^{2} - 4 z$

Step 3.5.16

Add $14 z^{2}$ and $4 z^{2}$ .

$18 z^{2} - 5 z^{4} - 8 - 4 z$

Step 3.6

Reorder terms.

$- 5 z^{4} + 18 z^{2} - 4 z - 8$