Question

attempt 6: 5 attempts remaining. let $f(x) = 9x^{1/5} - 8x^{2/7}$. evaluate each of the following: 1. $f(2) = $ 2. $f(5) = $ video example: solving a similar problem

Explanation:

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

We use the power rule for differentiation, which states that if \( y = ax^n \), then \( y' = anx^{n - 1} \).

For \( f(x)=9x^{\frac{1}{5}}-8x^{\frac{2}{7}} \), we differentiate term by term.

The derivative of \( 9x^{\frac{1}{5}} \) is \( 9\times\frac{1}{5}x^{\frac{1}{5}-1}=\frac{9}{5}x^{-\frac{4}{5}} \)

The derivative of \( - 8x^{\frac{2}{7}} \) is \( - 8\times\frac{2}{7}x^{\frac{2}{7}-1}=-\frac{16}{7}x^{-\frac{5}{7}} \)

So \( f^{\prime}(x)=\frac{9}{5}x^{-\frac{4}{5}}-\frac{16}{7}x^{-\frac{5}{7}} \)

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

Substitute \( x = 2 \) into \( f^{\prime}(x) \):

\( f^{\prime}(2)=\frac{9}{5}(2)^{-\frac{4}{5}}-\frac{16}{7}(2)^{-\frac{5}{7}} \)

\( (2)^{-\frac{4}{5}}=\frac{1}{2^{\frac{4}{5}}}=\frac{1}{\sqrt[5]{2^{4}}}=\frac{1}{\sqrt[5]{16}} \)

\( (2)^{-\frac{5}{7}}=\frac{1}{2^{\frac{5}{7}}}=\frac{1}{\sqrt[7]{2^{5}}}=\frac{1}{\sqrt[7]{32}} \)

\( f^{\prime}(2)=\frac{9}{5\sqrt[5]{16}}-\frac{16}{7\sqrt[7]{32}} \)

We can also write it as \( f^{\prime}(2)=\frac{9}{5\times2^{\frac{4}{5}}}-\frac{16}{7\times2^{\frac{5}{7}}} \)

If we want a decimal approximation:

\( 2^{\frac{4}{5}}\approx2^{0.8}\approx1.7411 \)

\( 2^{\frac{5}{7}}\approx2^{0.7143}\approx1.6438 \)

\( \frac{9}{5\times1.7411}-\frac{16}{7\times1.6438}\approx\frac{9}{8.7055}-\frac{16}{11.5066}\approx1.0338 - 1.3905\approx - 0.3567 \)

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

Substitute \( x = 5 \) into \( f^{\prime}(x) \):

\( f^{\prime}(5)=\frac{9}{5}(5)^{-\frac{4}{5}}-\frac{16}{7}(5)^{-\frac{5}{7}} \)

\( (5)^{-\frac{4}{5}}=\frac{1}{5^{\frac{4}{5}}}=\frac{1}{\sqrt[5]{5^{4}}}=\frac{1}{\sqrt[5]{625}} \)

\( (5)^{-\frac{5}{7}}=\frac{1}{5^{\frac{5}{7}}}=\frac{1}{\sqrt[7]{5^{5}}}=\frac{1}{\sqrt[7]{3125}} \)

\( f^{\prime}(5)=\frac{9}{5\sqrt[5]{625}}-\frac{16}{7\sqrt[7]{3125}} \)

In decimal approximation:

\( 5^{\frac{4}{5}}\approx5^{0.8}\approx3.3113 \)

\( 5^{\frac{5}{7}}\approx5^{0.7143}\approx2.9512 \)

\( \frac{9}{5\times3.3113}-\frac{16}{7\times2.9512}\approx\frac{9}{16.5565}-\frac{16}{20.6584}\approx0.5436 - 0.7745\approx - 0.2309 \)

Answer:

1. \( f^{\prime}(2)=\frac{9}{5\times2^{\frac{4}{5}}}-\frac{16}{7\times2^{\frac{5}{7}}}\approx - 0.357 \) (or in exact form \( \frac{9}{5\sqrt[5]{16}}-\frac{16}{7\sqrt[7]{32}} \))
2. \( f^{\prime}(5)=\frac{9}{5\times5^{\frac{4}{5}}}-\frac{16}{7\times5^{\frac{5}{7}}}\approx - 0.231 \) (or in exact form \( \frac{9}{5\sqrt[5]{625}}-\frac{16}{7\sqrt[7]{3125}} \))