Question

Let f be a differentiable function such that \(f(1)=2\) and \(f'(x)=f(x)\) for all \(x\in R\). If \(h(x)=f(f(x))\), then \(h'(1)\) is equal to :

• A. \(4e\)
• B. \(2e^2\)
• C. \(4e^2\)
• D. \(2e\)

Hint:

Solve \(f'=f\) to get \(f(x)=2e^{x-1}\), then apply the chain rule: \(h'(1)=f'(f(1))f'(1)\).

Answer 1

The Correct Option is A

\(f'=f\Rightarrow f(x)=Ce^x\); \(f(1)=2\Rightarrow C=2/e\), so \(f(x)=2e^{x-1}\).
Chain rule: \(h'(x)=f'(f(x))\cdot f'(x)\), so \(h'(1)=f'(f(1))\,f'(1)=f'(2)\,f'(1)\).
Since \(f'=f\): \(f'(2)=f(2)=2e\) and \(f'(1)=f(1)=2\). Therefore \(h'(1)=2e\cdot2=4e\).
Answer: (1) \(4e\)