Question

If \[y=f(x)\] is a differentiable function of \[x\] and the inverse function \[x=f^{-1}(y)\] exists, which expression gives \[\frac{\mathrm{d}x}{\mathrm{d}y}\]?

Options

• \[\frac{\mathrm{d}x}{\mathrm{d}y}=-\frac{1}{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}\]

• \[\frac{\mathrm{d}x}{\mathrm{d}y}=\frac{1}{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}\]

• \[\frac{\mathrm{d}x}{\mathrm{d}y}=\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2\]

• \[\frac{\mathrm{d}x}{\mathrm{d}y}=\frac{\mathrm{d}y}{\mathrm{d}x}\]

Answer 1

The derivative of an inverse function is the reciprocal of the derivative of the original function. This relation applies when \[\frac{\mathrm{d}y}{\mathrm{d}x}\neq0\].