Question

Q 24, function(2)

$$\begin{gathered} \mathbf{*}\mathbf{24}\mathbf{.}Ifgistheinverseofafunctionfandf\text{'}\left(x\right)=\frac{5}{1+x^5},theng\text{'}\left(x\right)isequalto \\ \left(A\right)\frac{1}{1+{\left\{g\left(x\right)\right\}}^5}\left(B\right)1+{\left\{g\left(x\right)\right\}}^5\left(C\right)1+x^5\left(D\right)5x^4 \end{gathered}$$

Answer 1

Dear student,
$$\begin{gathered} Given: \\ gistheinverseofafuntionfandf\text{'}\left(x\right)=\frac{1}{1+x^5} \\ Ifgistheinverseofafuntionf.so,therewillbef\left[g\left(x\right)\right]=x. \\ \Rightarrow f\left[g\left(x\right)\right]=x \\ differentiatewithrespecttox \\ \Rightarrow f\text{'}\left[g\left(x\right)\right]g\text{'}\left(x\right)=1 \\ \Rightarrow g\text{'}\left(x\right)=\frac{1}{f\text{'}\left[g\left(x\right)\right]} \\ \Rightarrow g\text{'}\left(x\right)=\frac{1}{1+{\left[g\left(x\right)\right]}^5}answer \end{gathered}$$
Regards