Question

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

Explanation:

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

Use the power - rule $\frac{d}{dx}(ax^n)=nax^{n - 1}$. Given $f(x)=x^{\frac{8}{5}}-2x^{\frac{7}{7}}=x^{\frac{8}{5}}-2x$. Then $f'(x)=\frac{8}{5}x^{\frac{8}{5}-1}-2=\frac{8}{5}x^{\frac{3}{5}}-2$.

Step2: Evaluate $f'(2)$

Substitute $x = 2$ into $f'(x)$. $f'(2)=\frac{8}{5}\times2^{\frac{3}{5}}-2$.

Step3: Evaluate $f'(4)$

Substitute $x = 4$ into $f'(x)$. First, $4^{\frac{3}{5}}=(2^2)^{\frac{3}{5}}=2^{\frac{6}{5}}$. Then $f'(4)=\frac{8}{5}\times4^{\frac{3}{5}}-2=\frac{8}{5}\times2^{\frac{6}{5}}-2$.

Answer:

$f'(2)=\frac{8}{5}\times2^{\frac{3}{5}}-2$
$f'(4)=\frac{8}{5}\times2^{\frac{6}{5}}-2$