assignment 4: problem 4 (1 point)\nlet $f(x)=-4x^{5}sqrt{x}+\frac{-7}{x^{2}sqrt{x}}$\n$f(x)=square$\nnote…

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=-4x^{5}\sqrt{x}+\frac{-7}{x^{2}\sqrt{x}}$ as $f(x)=-4x^{5}x^{\frac{1}{2}}- 7x^{-2}x^{-\frac{1}{2}}=-4x^{\frac{10 + 1}{2}}-7x^{\frac{-4-1}{2}}=-4x^{\frac{11}{2}}-7x^{-\frac{5}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$. For $y=-4x^{\frac{11}{2}}$, $y^\prime=-4\times\frac{11}{2}x^{\frac{11}{2}-1}=-22x^{\frac{9}{2}}$. For $y = - 7x^{-\frac{5}{2}}$, $y^\prime=-7\times(-\frac{5}{2})x^{-\frac{5}{2}-1}=\frac{35}{2}x^{-\frac{7}{2}}$.

Step3: Combine the derivatives

$f^\prime(x)=-22x^{\frac{9}{2}}+\frac{35}{2}x^{-\frac{7}{2}}$.

Answer:

$-22x^{\frac{9}{2}}+\frac{35}{2}x^{-\frac{7}{2}}$