Inverse Functions

Excerpts

• Inverse Functions

  It is not necessary that we should be able to solve the equation $y = \phi x$ in finite terms, that is, so as to give a value of $x$ without infinite series; it is sufficient that $x$ can be so expressed that the value of $x$ corresponding to any value of $y$ may be found as near as we please from $x = \psi y$, in the same manner as the value of $y$ corresponding to any value of $x$ is found from $y = \phi x$.

• Inverse Functions

  That is, the effect of the operation or set of operations denoted by $\psi$ is destroyed by the effect of those denoted by $\phi$; as in the instances $(x^{2})^{\efrac{1}{2}}$, $(x^{3})^{\efrac{1}{3}}$, $e^{\log x}$, angle whose sine is $(\sin x)$, etc., each of which is equal to $x$.

• Inverse Functions

  Hence $\dfrac{dx}{dy}$ as deduced from the second, and $\dfrac{dy}{dx}$ as deduced from the first, are reciprocals for every value of $dx$.

• Inverse Functions

  There is no very obvious analogy between $\dfrac{d^{2} y}{dx^{2}}$ and $\dfrac{d^{2} x}{dy^{2}}$; indeed no such appears from the method in which these coefficients were first formed.

• Inverse Functions

  Therefore $dy$, $dy_{1}$, etc., are not equal; whence arises the next column of second differences, or $d^{2} y$, $d^{2} y_{1}$, etc.

• Inverse Functions

  The resulting values of $y$, or $y$, $y_{1}$, etc., are not equidistant, except in one function only, when $y = ax + b$, where $a$ and $b$ are constant.

• Inverse Functions

  The limiting ratio of $d^{2} y$ to $(dx)^{2}$, expressed by $\dfrac{d^{2} y}{dx^{2}}$, is the second differential coefficient of $y$ with respect to $x$.

Equations

• This equation is in The Drawing of a Tangent to a Curve (The Drawing of a Tangent to a Curve)

• Inverse Functions

  x = \psi y

  Solving the first relation for x gives x as another function psi of y, the inverse relation.

• This equation is in Algebraical Geometry (Algebraical Geometry)

• Inverse Functions

  x = y^{\efrac{1}{2}}

  The inverse of y = x squared is x equal to y to the power one half.

• Inverse Functions

  x = \psi(\phi x)

  Substituting y = phi x into x = psi y gives x = psi(phi x): psi composed with phi returns x, so the operations of psi undo those of phi.

• Inverse Functions

  \dfrac{dy}{dx} = \phi' x

  The derivative of y with respect to x equals phi prime of x, obtained by differentiating y = phi x.

• Inverse Functions

  \dfrac{dx}{dy} = \psi' y

  The derivative of x with respect to y equals psi prime of y, obtained by differentiating x = psi y.

• Inverse Functions

  \frac{dy}{dx} = \phi' x = \frac{1}{\psi' y} = \frac{1}{p}

  The derivative dy/dx equals phi prime of x, which is the reciprocal of psi prime of y, where p stands for psi prime of y.

• Inverse Functions

  u = \frac{1}{p}

  Auxiliary quantity u is defined as the reciprocal of p.

• Inverse Functions

  \frac{du}{dp} = -\frac{1}{p^{2}}

  The derivative of u = 1/p with respect to p is minus one over p squared.

• Inverse Functions

  p = \psi' y

  The quantity p is defined as psi prime of y.

• Inverse Functions

  \frac{dp}{dy} = \psi'' y

  The derivative of p with respect to y equals psi double prime of y, the second derivative of psi.

• Inverse Functions

  \dfrac{d^{2} y}{dx^{2}} = \phi'' x

  The second differential coefficient of y with respect to x equals phi double prime of x.

• Inverse Functions

  \dfrac{d^{2} x}{dy^{2}} = \psi'' x

  The second differential coefficient of x with respect to y is written as psi double prime of x. As printed, the variable should be y (psi double prime of y), since psi is applied to y; this is a candidate erratum to flag, not silently corrected.

• Inverse Functions

  y = ax + b

  The linear case: y is a linear function of x with constant a and b; only in this case are values of y equidistant when x is equidistant.

• Inverse Functions

  y = e^{x}

  Example: y equals e to the power x, whose inverse is x = log y.

• Inverse Functions

  x = \log y

  The inverse of y = e^x is x equal to the logarithm of y.

• Inverse Functions

  \dfrac{dy}{dx} = e^{x}

  For y = e^x, the derivative of y with respect to x is e^x.

• Inverse Functions

  \dfrac{dx}{dy} = \dfrac{1}{y}

  For x = log y, the derivative of x with respect to y is 1/y.

• Inverse Functions

  \dfrac{d^{2} y}{dx^{2}} = e^{x}

  For y = e^x, the second differential coefficient of y with respect to x is e^x.

• Inverse Functions

  \dfrac{d^{2} x}{dy^{2}} = -\dfrac{1}{y^{2}}

  For x = log y, the second differential coefficient of x with respect to y is minus one over y squared.

Problems

No exercises in this chapter.