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PARTIAL DIFFERENTIATION

177

The first is obtained by supposing $y$ constant, the second is obtained by supposing $x$ constant; then

$dw=(4ax+3by)\,dx+(3bx+12cy^{2})\,dy$ .

Example (2). Let $z=x^{y}$ . Then, treating first $y$ and then $x$ as constant, we get in the usual way

$\left.{\begin{aligned}{\dfrac {\partial z}{\partial x}}&=yx^{y-1},\\{\dfrac {\partial z}{\partial y}}&=x^{y}\times \log _{\epsilon }x,\end{aligned}}\right\}$

so that $dz=yx^{y-1}\,dx+x^{y}\log _{\epsilon }x\,dy$ .

Example (3). A cone having height $h$ and radius of base $r$ has volume $V={\frac {1}{3}}\pi r^{2}h$ . If its height remains constant, while $r$ changes, the ratio of change of volume, with respect to radius, is different from ratio of change of volume with respect to height which would occur if the height were varied and the radius kept constant, for

$\left.{\begin{aligned}{\frac {\partial V}{\partial r}}&={\dfrac {2\pi }{3}}rh,\\{\frac {\partial V}{\partial h}}&={\dfrac {\pi }{3}}r^{2}.\end{aligned}}\right\}$

The variation when both the radius and the height change is given by $dV={\dfrac {2\pi }{3}}rh\,dV+{\dfrac {\pi }{3}}r^{2}\,dh$ .

Example (4). In the following example $F$ and $f$ denote two arbitrary functions of any form whatsoever. For example, they may be sine-functions, or exponentials, or mere algebraic functions of the two

C.M.E.

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