Given a power function of the form u ( s ) = a s n , with u ′ ( 3 ) = 4 and u ′ ( 6 ) = 8 , find n and a .

Answer 1

The power function u (s) = a sⁿ has the derivative u' (s) = a n s^{n - 1}

Given: u' (3) = a n 3^{n - 1} = 4

u' (6) = a n 6^{n - 1} = 8

Then u' (6) / u' (3) = 8 / 4

(a n 6^{n - 1}) / (a n 3^{n - 1}) = 8 / 4

6^{n - 1} / 3^{n - 1} = 8 / 4

(6 / 3)^{n - 1} = 8 / 4

2^{n - 1} = 8 / 4 = 2¹

n - 1 = 1

n = 2

After we found n, we substitute its value into u' (3):

u' (3) = a (2) 3^{2 - 1} = 4

a (2) (3) = 4

a = 4 / 6 = 2/3

Finally, n = 2, a = 2/3.