Suppose 𝑓(6)=4, 𝑓′(6)=5, 𝑓(40)=6, 𝑓′(40)=7, 𝑔(6)=4, 𝑔′(6)=−9, and 𝐻(𝑥)=𝑓(𝑥2+𝑔(𝑥)). Find 𝑑𝐻𝑑𝑥∣∣∣∣𝑥=6. 𝐻′(6)=

Answer 1

H(x) = f(x² + g(x))

By the Chain Rule, H'(x) = f'(x² + g(x)) [x² + g(x)]' = f'(x² + g(x))[2x + g'(x)]

So, H'(6) = f'(36 + 4)[12 - 9] = f'(40)(3) = 21