Question

homework4: problem 11 (1 point) let $f(x)=\frac{5x^{3}-5x^{2}}{sqrt3{x^{2}}}$. evaluate $f(x)$ at $x = 1$. $f(1)=$

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{5x^{3}-5x^{2}}{\sqrt[3]{x^{2}}}=5x^{3 - \frac{2}{3}}-5x^{2-\frac{2}{3}}=5x^{\frac{7}{3}}-5x^{\frac{4}{3}}$.

Step2: Differentiate using power - rule

The power - rule for differentiation is $(x^n)'=nx^{n - 1}$. So, $f'(x)=5\times\frac{7}{3}x^{\frac{7}{3}-1}-5\times\frac{4}{3}x^{\frac{4}{3}-1}=\frac{35}{3}x^{\frac{4}{3}}-\frac{20}{3}x^{\frac{1}{3}}$.

Step3: Evaluate at $x = 1$

Substitute $x = 1$ into $f'(x)$. Since $1^n=1$ for any real number $n$, we have $f'(1)=\frac{35}{3}\times1^{\frac{4}{3}}-\frac{20}{3}\times1^{\frac{1}{3}}=\frac{35}{3}-\frac{20}{3}=\frac{35 - 20}{3}=5$.

Answer:

$5$