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}}]