Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/223

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But from (A), and that is,

(C)

Solving for [1] we get

(57)

a formula for differentiating implicit functions. This formula in the form (C) is equivalent to the process employed in §62, for differentiating implicit functions, and all the examples at the end of §63 may be solved by using formula (57). Since

(D)

for all admissible values of and , we may say that (57) gives the relative time rates of change of and which keep from changing at all. Geometrically this means that the point must move on the curve whose equation is (D), and (57) determines the direction of its motion at any instant. Since

we may write (57) in the form of

(57a)

Illustrative Example 1. Given , find .

Solution. Let

∴ from (57a), Ans.

Illustrative Example 2. If increases at the rate of 2 inches per second as it passes through the value inches, at what rate must change when inch, in order that the function shall remain constant?

Solution. Let ; then

Substituting in (57a),
or By (33),§94
But x = 3, y = 1, ft. per second. Ans.
  1. It is assumed that and exist.