Question

Let y =(root3(x**5)).

At x = 64 the slope of the curve = [exact]

At x = 64 the slope of the curve = [approximate]

Answer 1

\[y = \sqrt[3]{x^5} = x^{\frac{5}{3}}\]\[y' = \frac {d}{dx} x^{\frac{5}{3}} = \frac{5}{3} x^{\frac{2}{3}}\]

Answer 2

\[y' (64) = (\frac {5}{3}) 64^{\frac{2}{3}} = \frac {5}{3} *16 = \frac {80}{3}\]

Answer 3

If you need an approximate, you can take the slop of a secant line near the point (64, 1024)
\[m \approx \frac {y(65)-y(63)}{65-63}\approx26.666425\] This approximation is pretty close to the exact of 80/3.