What is g^{\prime}(0) equal to?
Direction: Consider the following for the two (02) items that follow : Let f:(-1,1)\rightarrow R be a differentiable function with f(0)=-1 and f^{\prime}(0)=1. Let h(x)=f(2f(x)+2) and g(x)=(h(x))^{2}.
- A. -4 ✓
- B. -2
- C. 0
- D. 4
Correct Answer: A. -4
Explanation
We have g(x) = (h(x))^2. By the chain rule, g'(x) = 2h(x)h'(x). We evaluate h(0) = f(2f(0)+2) = f(2(-1)+2) = f(0) = -1. From the previous question, h'(0) = 2. Therefore, g'(0) = 2(-1)(2) = -4.
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