Derivative rule problem

Question

Given a power function of the form u(s)=as^n with u'(2)=20 and u'(6)=540, find n and a.

Answer 1

u(s) = asⁿ

u'(s) = ans^n-1

Since u'(2) = 20, an(2)^n-1 = 20 So, an = 20 / (2)^n-1

Since u'(6) = 540, an(6)^n-1 = 540 So, an = 540 / (6)^n-1

20 / (2)^n-1 = 540 / (6)^n-1

20(6)^n-1 = 540(2)^n-1

6^n-1 / 2^n-1 = 27

(6/2)^n-1 = 27

3^n-1 = 3³ So, n - 1 = 3

n = 4

Since n = 4 and an = 20 / (2)^n-1, we have 4a = 20 / 8 = 5/2. So, a = 5/8.