Total and Partial Differential Coefficients. Implicit Differentiation

Excerpts

• Total and Partial Differential Coefficients. Implicit Differentiation

  Generally, the complete differential coefficient of $z$ with respect to $x$, will contain as many terms as there are different ways in which $z$ contains $x$. From looking at a complete differential coefficient, we may see in what manner the function contained its variable.

• Total and Partial Differential Coefficients. Implicit Differentiation

  Find the differential coefficient belonging to each of the ways in which $z$ will contain $x$, as if it were the only way; the sum of these results (with their proper signs) will be the total differential coefficient.

• Total and Partial Differential Coefficients. Implicit Differentiation

  In saying that $z$ is a function of $x$ and $y$, and that $y$ is a function of $x$, we have first supposed $x$ to vary, $y$ remaining the same. The student must not imagine that $y$ is *then* a function of $x$; for if so, it would vary when $x$ varied.

• Total and Partial Differential Coefficients. Implicit Differentiation

  let $z = \log(x^{2} + a^{2})$. If we make $y = x^{2} + a^{2}$, we have $z = \log y$, and $\dfrac{dz}{dy} = \dfrac{1}{y}$; while from the first equation $\dfrac{dy}{dx} = 2x$.

• Total and Partial Differential Coefficients. Implicit Differentiation

  If $z = \log\log\sin x$, or the logarithm of the logarithm of $\sin x$, let $\sin x = y$ and $\log y = a$; whence $z= \log a$, and contains $x$, because $a$ contains $y$, which contains $x$.

• Total and Partial Differential Coefficients. Implicit Differentiation

  When $x$ is contained in $y$, and $y$ is contained in $z$, we shall say that $z$ is an indirect function of $x$ *through* $y$.

Equations

• Total and Partial Differential Coefficients. Implicit Differentiation

  d.z = \frac{dz}{dx}\, dx + \frac{dz}{dy}\, p\, dx

  When y is itself a function of x, the increment of z is the x-part plus the y-part, with dy replaced by p dx, where p is the differential coefficient of y with respect to x.

• Total and Partial Differential Coefficients. Implicit Differentiation

  \frac{d.z}{dx} = \frac{dz}{dx} + \frac{dz}{dx}\, p

  The total differential coefficient of z with respect to x equals the partial coefficient plus the partial coefficient times p = dy/dx; as printed the second term reads dz/dx, though the surrounding text requires dz/dy (possible misprint in this edition, flagged for checking).

• Total and Partial Differential Coefficients. Implicit Differentiation

  d.z = \frac{dz}{dx}\, dx + \frac{dz}{dy}\, dy + \frac{dz}{da}\, da + \etc.

  With x, y and a varying independently, the increment of z is the sum of the partial increments with respect to each variable, continued for further variables.

• Total and Partial Differential Coefficients. Implicit Differentiation

  da = \frac{da}{dx}\, dx + \frac{da}{dy}\, dy

  Since a varies as a function of y and x, its increment is the sum of its partial increments in x and in y.

• Total and Partial Differential Coefficients. Implicit Differentiation

  \frac{d.z}{dx} = \frac{dz}{dx} + \frac{dz}{dy}\, \frac{dy}{dx} + \frac{dz}{da}\, \frac{da}{dy}\, \frac{dy}{dx} + \frac{dz}{da}\, \frac{da}{dx}

  The complete differential coefficient of z with respect to x is the sum over every way z contains x: direct, through y, through a and y, and through a directly.

• Total and Partial Differential Coefficients. Implicit Differentiation

  \frac{dz}{dx} = \frac{dz}{da}\, \frac{da}{dy}\, \frac{dy}{dx}

  For z = log a, a = log y, y = sin x, the derivative of z with respect to x is the product of the three successive coefficients.

• Total and Partial Differential Coefficients. Implicit Differentiation

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

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

• Total and Partial Differential Coefficients. Implicit Differentiation

  \frac{dz}{da} = \frac{1}{a}

  For z = log a, the derivative of z with respect to a is 1/a.

• Total and Partial Differential Coefficients. Implicit Differentiation

  \frac{da}{dy} = \frac{1}{y}

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

• Total and Partial Differential Coefficients. Implicit Differentiation

  \frac{dy}{dx} = \cos x

  For y = sin x, the derivative of y with respect to x is cos x.

Problems

No exercises in this chapter.