Calculus Examples

$f (z) = \frac{z^{2} + 1}{3 z}$

Step 1

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

$\frac{1}{3} \frac{d}{d z} [\frac{z^{2} + 1}{z}]$

Step 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 $f (z) = z^{2} + 1$ and $g (z) = z$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$\frac{1}{3} \cdot \frac{z (2 z + 0) - (z^{2} + 1) \frac{d}{d z} [z]}{z^{2}}$

Step 3.4

Add $2 z$ and $0$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

Combine fractions.

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Step 9.1

Multiply $- 1$ by $1$ .

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

Step 9.2

Multiply $\frac{1}{3}$ by $\frac{2 z^{2} - (z^{2} + 1)}{z^{2}}$ .

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

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{2 z^{2} - z^{2} - 1 \cdot 1}{3 z^{2}}$

Step 10.2

Simplify the numerator.

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Step 10.2.1

Multiply $- 1$ by $1$ .

$\frac{2 z^{2} - z^{2} - 1}{3 z^{2}}$

Step 10.2.2

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

$\frac{z^{2} - 1}{3 z^{2}}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

Rewrite $1$ as $1^{2}$ .

$\frac{z^{2} - 1^{2}}{3 z^{2}}$

Step 10.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 1$ .

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