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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Differentiation

NDA 2024 · Mathematics · Differentiation

Let f(x) and g(x) be two functions such that g(x)=x-(1)/(x) and f o...

Let f(x)f(x) and g(x)g(x) be two functions such that g(x)=x−1xg(x)=x-\frac{1}{x} and ff o g(x)=x3−1x3g(x)=x^3-\frac{1}{x^3}. What is f′(x)f^{\prime}(x) equal to?

Options

  1. A−2x3-\frac{2}{x^3}
  2. B2x+2x32 x+\frac{2}{x^3}
  3. C6x+36 x+3
  4. D6x6 x

Correct answer

D. 6x6 x

Step-by-step solution

∵f(x)\because f(x) and g(x) are two functions g(x)=x−1x and fog(x)=x3−1x3f(x−1x)=x3−1x3⇒f(x−1x)=(x−1x)3+3(x−1x)\begin{aligned} g(x) & =x-\frac{1}{x} \text{ and } f o g(x)=x^3-\frac{1}{x^3} \\ f\left(x-\frac{1}{x}\right) & =x^3-\frac{1}{x^3} \\ \Rightarrow f\left(x-\frac{1}{x}\right) & =\left(x-\frac{1}{x}\right)^3+3\left(x-\frac{1}{x}\right) \end{aligned}f′(x)=3x2+3f′′(x)=6x\begin{aligned} f^{\prime}(x) & =3 x^2+3 \\ f^{\prime \prime}(x) & =6 x\end{aligned}$
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