Calculus Examples

$w = z^{5} - e$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d z} (w) = \frac{d}{d z} (z^{5} - e)$

Step 2

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

$w'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $z^{5} - e$ with respect to $z$ is $\frac{d}{d z} [z^{5}] + \frac{d}{d z} [- e]$ .

$\frac{d}{d z} [z^{5}] + \frac{d}{d z} [- e]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d z} [z^{n}]$ is $n z^{n - 1}$ where $n = 5$ .

$5 z^{4} + \frac{d}{d z} [- e]$

Step 3.3

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

$5 z^{4} + 0$

Step 3.4

Add $5 z^{4}$ and $0$ .

$5 z^{4}$

Step 4

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

$w' = 5 z^{4}$

Step 5

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

$\frac{d w}{d z} = 5 z^{4}$