Expert answers to "How do you find the instantaneous rate of..."

luminarc24lry

Answered question

2022-04-23

How do you find the instantaneous rate of change of w with respect to z for

w = \frac{1}{z} + \frac{z}{2}

Tyler Velasquez

Beginner2022-04-24Added 19 answers

\frac{d w}{d z} = \frac{d}{d z} (\frac{1}{z} + \frac{z}{2})

Initial set-up.

\frac{d w}{d z} = \frac{d}{d z} (\frac{1}{z}) + \frac{d}{d z} (\frac{z}{2})

The derivative of a sum is equal to the sum of the derivatives.

\frac{d w}{d z} = \frac{d}{d z} (z^{- 1}) + \frac{1}{2} \frac{d}{d z} (z)

A function

f (z) = \frac{c}{z^{n}}

with c constant can also be written as

f (z) = c z^{- n}

Second part:

\frac{d}{d z} c f (z) = c \frac{d}{d z} f (z)

if c is constant.

\frac{d w}{d z} = - 1 \cdot z^{- 2} + \frac{1}{2} \cdot 1

Use of the power rule:

\frac{d}{d z} z^{n} = n z^{n - 1}

. Then

\frac{d}{d z} z = \frac{d}{d z} z^{1} = z^{0} = 1

\frac{d w}{d z} = - z^{- 2} + \frac{1}{2}

Multiplicative identity postulate.

\frac{d w}{d z} = - \frac{1}{z^{2}} + \frac{1}{2}

A function written as

f (z) = c z^{- n}

can also be written

f (z) = \frac{c}{z^{n}}

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