consider the function h(x), for which h(3)=13 and h'(x)=7. find f'(3) for the function f(x)=h(x)/x.

Answer 1

There are two ways; I will show you both.

(I) By forming h(x)

From h'(x) = 7

h(x) = 7x + C for some constant C.

The constant C is found from the requirement that h(3) = 13:

13 = 7×3 + C

C = -8

So

h(x) = 7x - 8

f(x) = 7 - 8/x

f'(x) = 8/x²

f'(3) = 8/9

(II) Without forming h(x)

Differentiate f(x) = h(x)/x

f'(x) = h'(x)/x - h(x)/x²

Substitute given numbers

f'(3) = h'(3)/3 - h(3)/3² = 7/3 - 13/9 = 8/9