Question

let (f(x)=7x^{2/5}-4x^{3/7}). evaluate each of the following

1. (f(3)=)
2. (f(5)=)

Explanation:

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

Use the power - rule $\frac{d}{dx}(ax^n)=nax^{n - 1}$.
For $y = 7x^{\frac{2}{5}}-4x^{\frac{3}{7}}$, $y'=f'(x)=7\times\frac{2}{5}x^{\frac{2}{5}-1}-4\times\frac{3}{7}x^{\frac{3}{7}-1}=\frac{14}{5}x^{-\frac{3}{5}}-\frac{12}{7}x^{-\frac{4}{7}}$.

Step2: Evaluate $f'(3)$

Substitute $x = 3$ into $f'(x)$.
$f'(3)=\frac{14}{5}\times3^{-\frac{3}{5}}-\frac{12}{7}\times3^{-\frac{4}{7}}=\frac{14}{5\times3^{\frac{3}{5}}}-\frac{12}{7\times3^{\frac{4}{7}}}$.
Calculate $\frac{14}{5\times3^{\frac{3}{5}}}\approx\frac{14}{5\times1.93}\approx1.45$ and $\frac{12}{7\times3^{\frac{4}{7}}}\approx\frac{12}{7\times2.17}\approx0.79$.
$f'(3)\approx1.45 - 0.79 = 0.66$.

Step3: Evaluate $f'(5)$

Substitute $x = 5$ into $f'(x)$.
$f'(5)=\frac{14}{5}\times5^{-\frac{3}{5}}-\frac{12}{7}\times5^{-\frac{4}{7}}=\frac{14}{5\times5^{\frac{3}{5}}}-\frac{12}{7\times5^{\frac{4}{7}}}$.
Calculate $\frac{14}{5\times5^{\frac{3}{5}}}\approx\frac{14}{5\times2.63}\approx1.06$ and $\frac{12}{7\times5^{\frac{4}{7}}}\approx\frac{12}{7\times2.87}\approx0.60$.
$f'(5)\approx1.06 - 0.60 = 0.46$.

Answer:

1. $f'(3)\approx0.66$
2. $f'(5)\approx0.46$