# Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/217

Similarly, if we pass the plane BCD through P parallel to the YOZ-plane, its equation is $x = a,$

and for the section $DPI, \tfrac{\partial z}{\partial y}$ means the same as $\tfrac{dz}{dy}$. Hence

$\frac{\partial z}{\partial y} = \frac{dz}{dy} = -\tan MT'P =$ slope of section DI at P.

Illustrative Example 1. Given the ellipsoid $\frac{x^2}{24} + \frac{y^2}{12} + \frac{z^2}{6} = 1$; find the slope of the section of the ellipsoid made (a) by the plane $y = 1$ at the point where $x = 4$ and $z$ is positive; (b) by the plane $x = 2$ at the point where $y = 3$ and $z$ is positive.

 Solution. Considering $y$ as constant, $\tfrac{2x}{24} + \tfrac{2z}{6} \tfrac{\partial z}{\partial x}$ = 0, or $\tfrac{\partial z}{\partial x} = -\tfrac{x}{4z}.$ When $x$ is a constant $\tfrac{2y}{12} + \tfrac{2z}{6} \tfrac{\partial z}{\partial y}$ = 0, or $\tfrac{\partial z}{\partial y} = -\tfrac{y}{2z}.$
(a) When $y = 1$ and $x = 4, z = \sqrt{\tfrac{3}{2}}$. ∴ $\tfrac{\partial z}{\partial x} = -\sqrt{\tfrac{2}{3}}.$ Ans.
(b) When $x = 2$ and $y = 3, z = \tfrac{1}{\sqrt{2}}$. ∴ $\tfrac{\partial z}{\partial y} = -\tfrac{3}{2} \sqrt{2}.$ Ans.
EXAMPLES
 1. $u = x^3 + 3 x^2 y - y^3.$ Ans. $\tfrac{\partial u}{\partial x} = 3 x^2 + 6xy;$ $\tfrac{\partial u}{\partial y} = 3 x^2 + 3 y^2.$ 2. $u = Ax^2 + Bxy + Cy^2 + Dx + Ey + F.$ $\tfrac{\partial u}{\partial x} = 2 Ax + By + d;$ $\tfrac{\partial u}{\partial y} = Bx + 2Cy + E.$ 3. $u = (ax^2 + by^2 + cz^2)^n.$ $\tfrac{\partial u}{\partial x} = \frac{2anxu}{ax^2 + by^2 + cz^2};$ $\tfrac{\partial u}{\partial y} = \frac{2bnxu}{ax^2 + by^2 + cz^2}$ 4. $u = \arcsin \tfrac{x}{y}.$ $\tfrac{\partial u}{\partial x} = \frac{1}{\sqrt{y^2 - x^2}};$ $\tfrac{\partial u}{\partial y} = -\frac{x}{y\sqrt{y^2 - x^2}}.$ 5. $u = x^y.$ $\tfrac{\partial u}{\partial x} = yx^{y - 1};$ $\tfrac{\partial u}{\partial y} = x^y \log x.$ 6. $u = ax^3y^2z + bxy^3z^4 + cy^6 + dxz^3.$ $\tfrac{\partial u}{\partial x} = 3ax^2y^2z + by^3z^4 + dz^3.$ $\tfrac{\partial u}{\partial y} = 2ax^3yz + 3bxy^2z^4 + 6cy^5.$ $\tfrac{\partial u}{\partial z} = ax^3y^2 + 4bxy^3z^3 + 3dxz^2.$