Final answer:
To find the value of H'(4) for H(x) = F(G(x)), the chain rule is applied, resulting in 54. To find H(4) when H(x) = G(F(x)), the composition gives a value of 4. Using the chain rule again for finding H'(4) when H(x) = G(F(x)), the result is 77.
Step-by-step explanation:
The student is asking to find values related to the function H(x), which is a composition of functions F(x) and G(x). We are given specific values for F and G at certain points, as well as their derivatives at those points.
b) H'(x) if H(x) = F(G(x))
To find H'(4), we use the chain rule. Since H(x) = F(G(x)), H'(x) equals F'(G(x)) × G'(x). First, we find G(4), which is given as 2, and F'(G(4)) = F'(2), which is given as 9. Then, we find G'(4), which is given as 6. Therefore, H'(4) = F'(G(4)) × G'(4) = F'(2) × G'(4) = 9 × 6 = 54.
c) H(4) and d) H'(4) if H(x) = G(F(x))
For part c), finding H(4) involves computing G(F(4)). Since F(4) = 3 and G(3) = 4, then H(4) = G(F(4)) = G(3) = 4.
To calculate H'(4) for part d), we again apply the chain rule. H'(x) equals G'(F(x)) × F'(x). Given F(4) = 3 and G'(F(4)) = G'(3) = 11 also F'(4) = 7, we find that H'(4) = G'(F(4)) × F'(4) = G'(3) × F'(4) = 11 × 7 = 77.