Mathematics - Differentiation Question with Solution | TestHub

We have, $e^{-x}f(x)=2+{\int }_0^x\sqrt{t^4+1}dtx\in (-1,1)$ On differentiating w.r.t. $x$, we get $$\begin{array}{ll}& e^{-x}\left(f^{\prime }(x)-f(x)\right)=\sqrt{x^4+1}\\ \Rightarrow & f^{\prime }(x)=f(x)+\sqrt{x^4+1}{e}^x\\ \because & f^{-1}\text{ is the inverse of }f\\ \therefore & f^{-1}(f(x))=x\\ \Rightarrow & f^{-1^{\prime }}(f(x))f^{\prime }(x)=1\\ \Rightarrow & f^{-1^{\prime }}(f(x))=\frac{1}{f^{\prime }(x)}\\ \Rightarrow & f^{-1^{\prime }}(f(x))=\frac{1}{f(x)+\sqrt{x^4+1}{e}^x}\\ \text{ At }& x=0,f(x)=2\\ & f^{-1^{\prime }}(2)=\frac{1}{2+1}=\frac{1}{3}\end{array}$$