Question

find the derivative of $f(x)=sqrt{x}(5x^{3}+x^{2}-4x - 9)$. $f(x)=$

Explanation:

Step1: Rewrite the function

Rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$, so $f(x)=x^{\frac{1}{2}}(5x^{3}+x^{2}-4x - 9)=5x^{\frac{7}{2}}+x^{\frac{5}{2}}-4x^{\frac{3}{2}}-9x^{\frac{1}{2}}$.

Step2: Apply power - rule for derivatives

The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$.
For $y = 5x^{\frac{7}{2}}$, $y^\prime=5\times\frac{7}{2}x^{\frac{7}{2}-1}=\frac{35}{2}x^{\frac{5}{2}}$.
For $y = x^{\frac{5}{2}}$, $y^\prime=\frac{5}{2}x^{\frac{5}{2}-1}=\frac{5}{2}x^{\frac{3}{2}}$.
For $y=-4x^{\frac{3}{2}}$, $y^\prime=-4\times\frac{3}{2}x^{\frac{3}{2}-1}=-6x^{\frac{1}{2}}$.
For $y = - 9x^{\frac{1}{2}}$, $y^\prime=-9\times\frac{1}{2}x^{\frac{1}{2}-1}=-\frac{9}{2}x^{-\frac{1}{2}}$.

Step3: Combine the derivatives

$f^\prime(x)=\frac{35}{2}x^{\frac{5}{2}}+\frac{5}{2}x^{\frac{3}{2}}-6x^{\frac{1}{2}}-\frac{9}{2}x^{-\frac{1}{2}}$.

Answer:

$\frac{35}{2}x^{\frac{5}{2}}+\frac{5}{2}x^{\frac{3}{2}}-6x^{\frac{1}{2}}-\frac{9}{2}x^{-\frac{1}{2}}$