Let f(x) = x - 5 . If g(x) = [f(4h(x) + 3)]^2 and h(1) = 4, h'(1) = -2 , then g'(1) = .....

Step-by-step solution

Given f(x) = x - 5 and g(x) = [f(4h(x) + 3)]^2 . Substituting the expression for f(x) into g(x) : g(x) = [(4h(x) + 3) - 5]^2 = [4h(x) - 2]^2 Differentiating both sides with respect to x using the chain rule: g'(x) = 2[4h(x) - 2] d dx [4h(x) - 2] g'(x) = 2[4h(x) - 2] 4h'(x) = 8[4h(x) - 2]h'(x) Substitute x = 1 into the derivative: g'(1) = 8[4h(1) - 2]h'(1) Given h(1) = 4 and h'(1) = -2 , we get: g'(1) = 8[4(4) - 2](-2) g'(1) = 8[16 - 2](-2) g'(1) = 8[14](-2) = -224 Answer: -224