Question

suppose that $f(x)=1x^{3/5}-2x^{2/7}$. evaluate each of the following:
$f(2)=
$f(4)=
note: you can earn partial credit on this problem.

Explanation:

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

Use the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$.
For $y = x^{\frac{3}{5}}$, $y^\prime=\frac{3}{5}x^{\frac{3}{5}-1}=\frac{3}{5}x^{-\frac{2}{5}}$; for $y = 2x^{\frac{2}{7}}$, $y^\prime=2\times\frac{2}{7}x^{\frac{2}{7}-1}=\frac{4}{7}x^{-\frac{5}{7}}$.
So $f^\prime(x)=\frac{3}{5}x^{-\frac{2}{5}}-\frac{4}{7}x^{-\frac{5}{7}}$.

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

Substitute $x = 2$ into $f^\prime(x)$:
$f^\prime(2)=\frac{3}{5}\times2^{-\frac{2}{5}}-\frac{4}{7}\times2^{-\frac{5}{7}}$
$=\frac{3}{5\times2^{\frac{2}{5}}}-\frac{4}{7\times2^{\frac{5}{7}}}$
$\approx\frac{3}{5\times1.32}- \frac{4}{7\times1.52}$
$=\frac{3}{6.6}-\frac{4}{10.64}$
$\approx0.4545 - 0.376$
$= 0.0785$.

Step3: Evaluate $f^\prime(4)$

Substitute $x = 4$ into $f^\prime(x)$:
$f^\prime(4)=\frac{3}{5}\times4^{-\frac{2}{5}}-\frac{4}{7}\times4^{-\frac{5}{7}}$
$=\frac{3}{5\times4^{\frac{2}{5}}}-\frac{4}{7\times4^{\frac{5}{7}}}$
$\approx\frac{3}{5\times1.74}-\frac{4}{7\times2.29}$
$=\frac{3}{8.7}-\frac{4}{16.03}$
$\approx0.3448-0.2495$
$=0.0953$.

Answer:

$f^\prime(2)\approx0.0785$
$f^\prime(4)\approx0.0953$