Question

suppose that (f(x)=9x^{4/5}-6x^{3/7}). evaluate each of the following: (f(2)=)(f(5)=)

Explanation:

Step1: Find the derivative of \(f(x)\)

Use the power - rule \((x^n)^\prime=nx^{n - 1}\). If \(f(x)=9x^{\frac{4}{5}}-6x^{\frac{3}{7}}\), then \(f^\prime(x)=9\times\frac{4}{5}x^{\frac{4}{5}-1}-6\times\frac{3}{7}x^{\frac{3}{7}-1}\).
\[f^\prime(x)=\frac{36}{5}x^{-\frac{1}{5}}-\frac{18}{7}x^{-\frac{4}{7}}\]

Step2: Evaluate \(f^\prime(2)\)

Substitute \(x = 2\) into \(f^\prime(x)\):
\[f^\prime(2)=\frac{36}{5}\times2^{-\frac{1}{5}}-\frac{18}{7}\times2^{-\frac{4}{7}}\]
\[f^\prime(2)=\frac{36}{5\times2^{\frac{1}{5}}}-\frac{18}{7\times2^{\frac{4}{7}}}\]
\[f^\prime(2)=\frac{36}{5\times\sqrt[5]{2}}-\frac{18}{7\times\sqrt[7]{16}}\]

Step3: Evaluate \(f^\prime(5)\)

Substitute \(x = 5\) into \(f^\prime(x)\):
\[f^\prime(5)=\frac{36}{5}\times5^{-\frac{1}{5}}-\frac{18}{7}\times5^{-\frac{4}{7}}\]
\[f^\prime(5)=\frac{36}{5\times\sqrt[5]{5}}-\frac{18}{7\times\sqrt[7]{625}}\]

Answer:

\[f^\prime(2)=\frac{36}{5\times2^{\frac{1}{5}}}-\frac{18}{7\times2^{\frac{4}{7}}}, f^\prime(5)=\frac{36}{5\times\sqrt[5]{5}}-\frac{18}{7\times\sqrt[7]{625}}\]