Question

if $f(x)=\frac{7x^{2}+6x + 23}{sqrt{x}}$, find $f(x)$. video example: solving a similar problem

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{7x^{2}+6x + 23}{\sqrt{x}}$ as $f(x)=7x^{\frac{3}{2}}+6x^{\frac{1}{2}}+23x^{-\frac{1}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$.
For $y = 7x^{\frac{3}{2}}$, $y^\prime=7\times\frac{3}{2}x^{\frac{3}{2}-1}=\frac{21}{2}x^{\frac{1}{2}}$.
For $y = 6x^{\frac{1}{2}}$, $y^\prime=6\times\frac{1}{2}x^{\frac{1}{2}-1}=3x^{-\frac{1}{2}}$.
For $y = 23x^{-\frac{1}{2}}$, $y^\prime=23\times(-\frac{1}{2})x^{-\frac{1}{2}-1}=-\frac{23}{2}x^{-\frac{3}{2}}$.

Step3: Combine the derivatives

$f^\prime(x)=\frac{21}{2}\sqrt{x}+\frac{3}{\sqrt{x}}-\frac{23}{2x\sqrt{x}}$.

Answer:

$f^\prime(x)=\frac{21}{2}\sqrt{x}+\frac{3}{\sqrt{x}}-\frac{23}{2x\sqrt{x}}$