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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Differentiation

NDA 2017 · Mathematics · Differentiation

Let f(x+y)=f(x) f(y) for all x and y. Then what is f^ (5) equal to...

Let f(x+y)=f(x)f(y)f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) f(\mathrm{y}) for all x and y. Then what is f′(5)f^{\prime}(5) equal to [where f′(x)f^{\prime}(\mathrm{x}) is the derivative of f(x)f(\mathrm{x}) ]?

Options

  1. Af(5)f′0f(5) f^{\prime} 0
  2. Bf(5)−f′(0)f(5)-f^{\prime}(0)
  3. Cf(5)f(0)f(5) f(0)
  4. Df(5)+f′(0)f(5)+f^{\prime}(0)

Correct answer

A. f(5)f′0f(5) f^{\prime} 0

Step-by-step solution

f(x+y)=f(x)⋅f(y)Let⁡f(x)=axf(x+y)=ax+y=ax⋅ay=f(x)⋅f(y)f(5)=a5f(5)=a5⋅log⁡a=f(5)⋅f(0)f1(0)=a0⋅log⁡a=log⁡a\begin{aligned} & f(x+y)=f(x) \cdot f(y) \\ & \operatorname{Let} f(x)=a^x \\ & f(x+y)=a^{x+y}=a^x \cdot a^y=f(x) \cdot f(y) \\ & f(5)=a^5 \\ & f(5)=a^5 \cdot \log a \\ & =f(5) \cdot f(0) \\ & f^1(0)=a^0 \cdot \log a \\ & =\log a\end{aligned}
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