What is the d/(dz)(1/(1-z^{-1)}) ?

The d/(dz)(1/(1-z^{-1)}) is -1/((z-1)^2)

What is the first d/(dz)(1/(1-z^{-1)}) ?

The first d/(dz)(1/(1-z^{-1)}) is -1/((z-1)^2)

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\frac{d}{d z} (\frac{1}{1 - z ^{- 1}})

- \frac{1}{( z - 1 ) ^{2}}

Solution steps

\frac{d}{d z} (\frac{1}{1 - z ^{- 1}})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

= - \frac{1}{( 1 - z ^{- 1} ) ^{2}} \cdot \frac{1}{z ^{2}}

Simplify

- \frac{1}{( 1 - z ^{- 1} ) ^{2}} \cdot \frac{1}{z ^{2}} : - \frac{1}{( z - 1 ) ^{2}}

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

What is the d/(dz)(1/(1-z^{-1)}) ?
The d/(dz)(1/(1-z^{-1)}) is -1/((z-1)^2)
What is the first d/(dz)(1/(1-z^{-1)}) ?
The first d/(dz)(1/(1-z^{-1)}) is -1/((z-1)^2)