Question

1. differentiate $f(z)=\frac{z^{8}+2}{sqrt{z}}$. $f(z)=$

Explanation:

Step1: Rewrite the function

Rewrite $f(z)=\frac{z^{8}+2}{\sqrt{z}}$ as $f(z)=z^{8 - \frac{1}{2}}+2z^{-\frac{1}{2}}=z^{\frac{15}{2}}+2z^{-\frac{1}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$.
For the first term $y_1 = z^{\frac{15}{2}}$, its derivative $y_1^\prime=\frac{15}{2}z^{\frac{15}{2}-1}=\frac{15}{2}z^{\frac{13}{2}}$.
For the second term $y_2 = 2z^{-\frac{1}{2}}$, its derivative $y_2^\prime=2\times(-\frac{1}{2})z^{-\frac{1}{2}-1}=-z^{-\frac{3}{2}}$.

Step3: Combine the derivatives

$f^\prime(z)=y_1^\prime + y_2^\prime=\frac{15}{2}z^{\frac{13}{2}}-z^{-\frac{3}{2}}$.

Answer:

$\frac{15}{2}z^{\frac{13}{2}}-z^{-\frac{3}{2}}$