(18) Suppose ​f(3) = 8, f'(3) = 3​, ​g(3) = 2, and g'(3) = 6. Find h(3) and ​h'(3), where h(x) = 4f(x) + 5g(x).

Question

h(3) = _________ (Simplify Your Answer).

Answer 1

Givens:

f(3) = 8

g(3) = 2

f'(3) = 3

g'(3) = 6

we know that:

h(x)=4f(x) + 5g(x)

we can plug in f(3) and g(3) directly

h(3) = 4f(3) + 5g(3) = 4(8) + 5(2) = 32 +10 = 42

With h'(x) we need to use the fact that derivatives are linear:

d(h(x))/dx = d/dx (4f(x) + 5g(x)) = d/dx (4f(x)) + d/dx(5g(x)) = 4f'(x) + 5g'(x)

h'(x) = 4f'(3) + 5g'(3) = 4(3) + 5(6) = 12 + 30 = 42