Calculus Examples

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

Step 1

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

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

Step 2

Evaluate $\frac{d}{d z} [z^{2} (z + 4)]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$z^{2} (1 + 0) + (z + 4) \frac{d}{d z} [z^{2}] + \frac{d}{d z} [- \frac{2 z}{z^{2} + 1}]$

Step 2.5

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

$z^{2} (1 + 0) + (z + 4) (2 z) + \frac{d}{d z} [- \frac{2 z}{z^{2} + 1}]$

Step 2.6

Add $1$ and $0$ .

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

Step 2.7

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

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

Step 2.8

Move $2$ to the left of $z + 4$ .

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

Step 3

Evaluate $\frac{d}{d z} [- \frac{2 z}{z^{2} + 1}]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

$z^{2} + 2 (z + 4) z - 2 \frac{(z^{2} + 1) \cdot 1 - z (2 z + 0)}{{(z^{2} + 1)}^{2}}$

Step 3.7

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

$z^{2} + 2 (z + 4) z - 2 \frac{z^{2} + 1 - z (2 z + 0)}{{(z^{2} + 1)}^{2}}$

Step 3.8

Add $2 z$ and $0$ .

$z^{2} + 2 (z + 4) z - 2 \frac{z^{2} + 1 - z (2 z)}{{(z^{2} + 1)}^{2}}$

Step 3.9

Multiply $2$ by $- 1$ .

$z^{2} + 2 (z + 4) z - 2 \frac{z^{2} + 1 - 2 z \cdot z}{{(z^{2} + 1)}^{2}}$

Step 3.10

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

$z^{2} + 2 (z + 4) z - 2 \frac{z^{2} + 1 - 2 (z^{1} z)}{{(z^{2} + 1)}^{2}}$

Step 3.11

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

$z^{2} + 2 (z + 4) z - 2 \frac{z^{2} + 1 - 2 (z^{1} z^{1})}{{(z^{2} + 1)}^{2}}$

Step 3.12

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

$z^{2} + 2 (z + 4) z - 2 \frac{z^{2} + 1 - 2 z^{1 + 1}}{{(z^{2} + 1)}^{2}}$

Step 3.13

Add $1$ and $1$ .

$z^{2} + 2 (z + 4) z - 2 \frac{z^{2} + 1 - 2 z^{2}}{{(z^{2} + 1)}^{2}}$

Step 3.14

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

$z^{2} + 2 (z + 4) z - 2 \frac{- z^{2} + 1}{{(z^{2} + 1)}^{2}}$

Step 3.15

Combine $- 2$ and $\frac{- z^{2} + 1}{{(z^{2} + 1)}^{2}}$ .

$z^{2} + 2 (z + 4) z + \frac{- 2 (- z^{2} + 1)}{{(z^{2} + 1)}^{2}}$

Step 3.16

Move the negative in front of the fraction.

$z^{2} + 2 (z + 4) z - \frac{2 (- z^{2} + 1)}{{(z^{2} + 1)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$z^{2} + (2 z + 2 \cdot 4) z - \frac{2 (- z^{2} + 1)}{{(z^{2} + 1)}^{2}}$

Step 4.2

Apply the distributive property.

$z^{2} + 2 z \cdot z + 2 \cdot 4 z - \frac{2 (- z^{2} + 1)}{{(z^{2} + 1)}^{2}}$

Step 4.3

Apply the distributive property.

$z^{2} + 2 z \cdot z + 2 \cdot 4 z - \frac{2 (- z^{2}) + 2 \cdot 1}{{(z^{2} + 1)}^{2}}$

Step 4.4

Combine terms.

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Step 4.4.1

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

$z^{2} + 2 (z^{1} z) + 2 \cdot 4 z - \frac{2 (- z^{2}) + 2 \cdot 1}{{(z^{2} + 1)}^{2}}$

Step 4.4.2

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

$z^{2} + 2 (z^{1} z^{1}) + 2 \cdot 4 z - \frac{2 (- z^{2}) + 2 \cdot 1}{{(z^{2} + 1)}^{2}}$

Step 4.4.3

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

$z^{2} + 2 z^{1 + 1} + 2 \cdot 4 z - \frac{2 (- z^{2}) + 2 \cdot 1}{{(z^{2} + 1)}^{2}}$

Step 4.4.4

Add $1$ and $1$ .

$z^{2} + 2 z^{2} + 2 \cdot 4 z - \frac{2 (- z^{2}) + 2 \cdot 1}{{(z^{2} + 1)}^{2}}$

Step 4.4.5

Multiply $2$ by $4$ .

$z^{2} + 2 z^{2} + 8 z - \frac{2 (- z^{2}) + 2 \cdot 1}{{(z^{2} + 1)}^{2}}$

Step 4.4.6

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

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

Step 4.4.7

Multiply $- 1$ by $2$ .

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

Step 4.4.8

Multiply $2$ by $1$ .

$3 z^{2} + 8 z - \frac{- 2 z^{2} + 2}{{(z^{2} + 1)}^{2}}$

Step 4.4.9

To write $3 z^{2}$ as a fraction with a common denominator, multiply by $\frac{{(z^{2} + 1)}^{2}}{{(z^{2} + 1)}^{2}}$ .

$8 z + \frac{3 z^{2} {(z^{2} + 1)}^{2}}{{(z^{2} + 1)}^{2}} - \frac{- 2 z^{2} + 2}{{(z^{2} + 1)}^{2}}$

Step 4.4.10

Combine the numerators over the common denominator.

$8 z + \frac{3 z^{2} {(z^{2} + 1)}^{2} - (- 2 z^{2} + 2)}{{(z^{2} + 1)}^{2}}$