Question

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

Explanation:

Step1: Rewrite the function

Rewrite $f(z)=\frac{z^{2}+1}{\sqrt{z}}$ as $f(z)=z^{\frac{3}{2}}+z^{-\frac{1}{2}}$ using the rule $\frac{a + b}{c}=\frac{a}{c}+\frac{b}{c}$ and $\frac{z^{n}}{\sqrt{z}}=z^{n-\frac{1}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = z^{n}$, then $y^\prime= nz^{n - 1}$.
For $y_1=z^{\frac{3}{2}}$, $y_1^\prime=\frac{3}{2}z^{\frac{3}{2}-1}=\frac{3}{2}z^{\frac{1}{2}}$.
For $y_2 = z^{-\frac{1}{2}}$, $y_2^\prime=-\frac{1}{2}z^{-\frac{1}{2}-1}=-\frac{1}{2}z^{-\frac{3}{2}}$.

Step3: Find the derivative of $f(z)$

$f^\prime(z)=y_1^\prime + y_2^\prime=\frac{3}{2}z^{\frac{1}{2}}-\frac{1}{2}z^{-\frac{3}{2}}$.

Answer:

$\frac{3}{2}z^{\frac{1}{2}}-\frac{1}{2}z^{-\frac{3}{2}}$