Calculus Examples

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

Step 1

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

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

Step 2

Evaluate $\frac{d}{d z} [3 z^{- 2}]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $- 2$ by $3$ .

$- 6 z^{- 3} + \frac{d}{d z} [- \frac{1}{z}]$

Step 3

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

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Step 3.1

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 $f (z) = - 1$ and $g (z) = \frac{1}{z}$ .

$- 6 z^{- 3} - \frac{d}{d z} [\frac{1}{z}] + \frac{1}{z} \frac{d}{d z} [- 1]$

Step 3.2

Rewrite $\frac{1}{z}$ as $z^{- 1}$ .

$- 6 z^{- 3} - \frac{d}{d z} [z^{- 1}] + \frac{1}{z} \frac{d}{d z} [- 1]$

Step 3.3

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

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

Step 3.4

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

$- 6 z^{- 3} - - z^{- 2} + \frac{1}{z} \cdot 0$

Step 3.5

Multiply $- 1$ by $- 1$ .

$- 6 z^{- 3} + 1 z^{- 2} + \frac{1}{z} \cdot 0$

Step 3.6

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

$- 6 z^{- 3} + z^{- 2} + \frac{1}{z} \cdot 0$

Step 3.7

Multiply $\frac{1}{z}$ by $0$ .

$- 6 z^{- 3} + z^{- 2} + 0$

Step 3.8

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

$- 6 z^{- 3} + z^{- 2}$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$z^{- 2} - 6 z^{- 3}$

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{z^{2}} - 6 z^{- 3}$

Step 4.2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{z^{2}} - 6 \frac{1}{z^{3}}$

Step 4.2.3

Combine $- 6$ and $\frac{1}{z^{3}}$ .

$\frac{1}{z^{2}} + \frac{- 6}{z^{3}}$

Step 4.2.4

Move the negative in front of the fraction.

$\frac{1}{z^{2}} - \frac{6}{z^{3}}$