Calculus Examples

$u = \frac{a z^{n} - 1}{c}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d z} (u) = \frac{d}{d z} (\frac{a z^{n} - 1}{c})$

Step 2

The derivative of $u$ with respect to $z$ is $u'$ .

$u'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$\frac{1}{c} \frac{d}{d z} [a z^{n} - 1]$

Step 3.2

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

$\frac{1}{c} (\frac{d}{d z} [a z^{n}] + \frac{d}{d z} [- 1])$

Step 3.3

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

$\frac{1}{c} (a \frac{d}{d z} [z^{n}] + \frac{d}{d z} [- 1])$

Step 3.4

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

$\frac{1}{c} (a (n z^{n - 1}) + \frac{d}{d z} [- 1])$

Step 3.5

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

$\frac{1}{c} (a n z^{n - 1} + 0)$

Step 3.6

Combine fractions.

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Step 3.6.1

Add $a n z^{n - 1}$ and $0$ .

$\frac{1}{c} (a n z^{n - 1})$

Step 3.6.2

Combine $a$ and $\frac{1}{c}$ .

$\frac{a}{c} (n z^{n - 1})$

Step 3.6.3

Combine $n$ and $\frac{a}{c}$ .

$\frac{n a}{c} z^{n - 1}$

Step 3.6.4

Combine $\frac{n a}{c}$ and $z^{n - 1}$ .

$\frac{n a z^{n - 1}}{c}$

Step 3.7

Simplify.

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Step 3.7.1

Reorder terms.

$\frac{z^{n - 1} a n}{c}$

Step 3.7.2

Reorder factors in $\frac{z^{n - 1} a n}{c}$ .

$\frac{a n z^{n - 1}}{c}$

Step 4

Reform the equation by setting the left side equal to the right side.

$u' = \frac{a n z^{n - 1}}{c}$

Step 5

Replace $u'$ with $\frac{d u}{d z}$ .

$\frac{d u}{d z} = \frac{a n z^{n - 1}}{c}$