Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 123410 by bemath last updated on 25/Nov/20

$$Supposethatfisdifferentiable$$$$withderivativef{}^{\prime }\left(x\right)={(1+x^3)}^{-1/2}.$$$$Showthatg=f^{-1}satisfies$$$$g^{″}\left(x\right)=\frac{3}{2}g{\left(x\right)}^2$$

Answered by mnjuly1970 last updated on 25/Nov/20

$$y=f\left(x\right)\Rightarrow x=f^{-1}\left(y\right)$$$${\left(f^{-1}\right)}^{{}^{\prime }}\left(y\right)=\frac{1}{f^{{}^{\prime }}\left(x\right)}\Rightarrow g^{\prime }\left(y\right)=\frac{1}{f{}^{\prime }\left(x\right)}$$$$g^{″}\left(y\right)y^{{}^{\prime }}=\frac{-f{}^{″}\left(x\right)}{{\left(f{}^{\prime }\right)}^2\left(x\right)}$$$$g^{{}^{″}}\left(y\right)=\frac{-f{}^{″}\left(x\right)}{{\left(f{}^{\prime }\right)}^3\left(x\right)}=\frac{-\frac{-1}{2}\left(3x^2\right){(1+x^3)}^{\frac{-3}{2}}}{{(1+x^3)}^{\frac{-3}{2}}}$$$$g^{″}\left(y\right)=\frac{3}{2}{x}^2=\frac{3}{2}{\left(f^{-1}\left(y\right)\right)}^2$$$$g^{″}\left(y\right)=\frac{3}{2}{g}^2\left(y\right)✓$$

Commented by bemath last updated on 25/Nov/20

Terms of Service

Privacy Policy

Contact: info@tinkutara.com