Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/224

 Let P be the point (x, y, z) on the surface given by the equation

and let PC and AP be sections made by planes through P parallel to the YOZ- and XOZ-planes respectively. Along the curve AP, y is constant; therefore, from (E), z is an implicit function of x alone, and we have, from (57a),



giving the slope at P of the curve AP, §122.

$$\tfrac{\partial z}{\partial x}$$ is used instead of $$\tfrac{dz}{dx}$$ in the first member, since z was originally, from (E), an implicit function of x and y; but (58) is deduced on the hypothesis that y remains constant.

Similarly, the slope at P of the curve PC is

Find the total derivatives, using (51), (52), or (53), in the following six examples: