Suppose 𝑓(5)=7, 𝑓′(5)=2, 𝑓(19)=2, 𝑓′(19)=−3, and 𝐻(𝑥)=𝑓(𝑥2−6). Find 𝑑𝐻𝑑𝑥∣∣∣∣𝑥=5. 𝐻′(5)=

Answer 1

The Chain Rule: H = f(g(x)), then H' = f'(g(x) . g'(x)

g(x) = x2-6

H'(x) = f'(x2-6) * g'(x)

H(5) = f'(19) * (2x) = -3* 2* 5 = -30.