Question

How do you find the instantaneous rate of change of `w` with respect to `z` for `w=1/z+z/2`?

Answer 1

`(dw)/dz = -1/z^2 + 1/2`

Explanation :

`(dw)/dz = d/dz (1/z + z/2)`

Initial set-up.

`(dw)/dz = d / dz ( 1/z) + d /dz ( z/2)`

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

`(dw)/dz = d/dz (z^-1) + 1/2 d/dz (z)`

First part: A function `f(z) = c/(z^n)` with `c` constant can also be written as `f(z) = cz^(-n)` Second part: `d/dz cf(z) = c d/dz f(z)` if c is constant.

`(dw)/dz = -1*z^ -2 + 1/2*1`

Use of the power rule: `d/dz z^n = nz^(n-1)`. Then `d/dz z = d/dz z^1 = z^0 = 1`

`(dw)/dz = -z^ -2 + 1/2`

Multiplicative identity postulate.

`(dw)/dz = -1/z^2 + 1/2`

A function written as `f(z) = cz^(-n)` can also be written `f(z) = c/(z^n)`