Question

3-34 Differentiate the function.

10. \(r\left( z \right) = {z^{ - 5}} - {z^{\frac{1}{2}}}\)

Short answer

The differentiation of the function \(r\left( z \right) = {z^{ - 5}} - {z^{\frac{1}{2}}}\) is \(r'\left( z \right) = - 5{z^{ - 6}} - \frac{1}{2}{z^{ - \frac{1}{2}}}\).

Step-by-step solution

Write the formula of the difference rule and the power rule

The Difference Rule:\(\frac{d}{{dx}}\left( {f\left( x \right) - g\left( x \right)} \right) = \frac{d}{{dx}}f\left( x \right) - \frac{d}{{dx}}g\left( x \right)\)

The Power Rule: \(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\) where \(n\) is a positive integer.

Find the differentiation of the function

Consider the function \(r\left( z \right) = {z^{ - 5}} - {z^{\frac{1}{2}}}\). Differentiate the function w.r.t \(z\) by using the difference rule and the power rule.

\(\begin{aligned}\frac{{d\left( {r\left( z \right)} \right)}}{{dz}} &= \frac{{d\left( {{z^{ - 5}} - {z^{\frac{1}{2}}}} \right)}}{{dz}}\\ &= \frac{d}{{dz}}\left( {{z^{ - 5}}} \right) + \frac{d}{{dz}}\left( { - {z^{\frac{1}{2}}}} \right)\\ &= \frac{d}{{dz}}\left( { - 5{z^{ - 5 - 1}}} \right) + \frac{d}{{dz}}\left( { - \frac{1}{2}{z^{\frac{1}{2} - 1}}} \right)\\ &= - 5{z^{ - 6}} - \frac{1}{2}{z^{ - \frac{1}{2}}}\end{aligned}\)

Thus, the derivative of the function \(r\left( z \right) = {z^{ - 5}} - {z^{\frac{1}{2}}}\) is \(r'\left( z \right) = - 5{z^{ - 6}} - \frac{1}{2}{z^{ - \frac{1}{2}}}\).