if f(x)=(x+2)/(x+3) and y=f^-1(x) then dy/dx is equal to

Question : If f(x)=(x+2)/(x+3) and y=f^-1(x) then dy/dx is equal to

Solution :

Let the Given function \[z=\frac{x+2}{x+3}\] \[xz+3z=x+2\] \[xz-x=2-3z\] x=\frac{2-3z}{z-1} This is inverse of f in terms of the variable z. By replacing z by x, we shall obtain inverse of f(x) in terms of the variable x c \[y=f^{-1}\left(x\right)=\frac{2-3x}{x-1}\] Differentiating with respect to x \[\frac{dy}{dx}=\frac{\left(x-1\right)\left(-3\right)-\left(2-3x\right)\left(1\right)}{\left(x-1\right)^2}\] \[\frac{dy}{dx}=\frac{-3x+3-2+3x}{\left(x-1\right)^2}\] \[\frac{dy}{dx}=\frac{1}{\left(x-1\right)^2}\] is the required derivative of f^-1(x).