Question

Differentiate the function.

\(F\left( z \right) = \frac{{A + Bz + C{z^2}}}{{{Z^2}}}\)

Short answer

Derivative of above function is \( - 2A{z^{ - 3}} - B{z^{ - 2}}\).

Step-by-step solution

Precise Definition of differentiation

A derivative is a relationship between changing of two variables. Consider independent variable x and dependent variable y. The change of dependent variable with respect to change of independent variable expression can be found by using the derivative formula.

Rewrite the function

Rewrite the given function.

\(\begin{aligned}F\left( z \right) &= \frac{{A + Bz + C{z^2}}}{{{z^2}}}\\ &= \frac{A}{{{z^2}}} + \frac{{Bz}}{{{z^2}}} + \frac{{C{z^2}}}{{{z^2}}}\\ &= A{z^{ - 2}} + B{z^{ - 1}} + C\end{aligned}\)

Power rule of differentiation

The power rule of differentiation is that \(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\), where \(n\) is any real number.

Apply power rule and simplify.

\(\begin{aligned}F'\left( z \right) &= \frac{d}{{dz}}\left( {A{z^{ - 2}} + B{z^{ - 1}} + C} \right)\\ &= \frac{d}{{dz}}\left( {A{z^{ - 2}}} \right) + \frac{d}{{dz}}\left( {B{z^{ - 1}}} \right) + \frac{d}{{dz}}\left( C \right)\\ &= A\left( { - 2{z^{ - 2 - 1}}} \right) + B\left( { - 1{z^{ - 1 - 1}}} \right) + 0\\ &= - 2A{z^{ - 3}} - B{z^{ - 2}}\end{aligned}\)

So, the differentiation of the function is \(F'(z) = - 2A{z^{ - 3}} - B{z^{ - 2}}\).