A) \[3{{x}^{2}}+3\]
B) \[{{x}^{2}}-\frac{1}{{{x}^{2}}}\]
C) \[1+\frac{1}{{{x}^{2}}}\]
D) \[3{{x}^{2}}+\frac{3}{{{x}^{4}}}\]
Correct Answer: A
Solution :
Given that, \[g(x)=x-\frac{1}{x}\] and \[fog(x)={{x}^{3}}-\frac{1}{{{x}^{3}}}\] \[\Rightarrow \] \[f(g(x)={{x}^{3}}-\frac{1}{{{x}^{3}}}\] \[\Rightarrow \] \[f\left( x-\frac{1}{x} \right)={{x}^{3}}-\frac{1}{{{x}^{3}}}\] \[=\left( x-\frac{1}{x} \right)\left( {{x}^{2}}+\frac{1}{{{x}^{2}}}+1 \right)\] \[\Rightarrow \] \[f\left( x-\frac{1}{x} \right)=\left( x-\frac{1}{x} \right)\left[ {{\left( x-\frac{1}{x} \right)}^{2}}+3 \right]\] \[\Rightarrow \] \[f(x)={{x}^{3}}+3x\] On differentiating w.r.t.\[x,\]we get
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