By Robert Carl Yates
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Extra resources for Analytic geometry with calculus
We shall make free use of this fact but leave its proof to a course in the Calculus. Illustrations. (A) y = (x2 + 3x - 1) (x4 + 7x + 5). [Form: y = uv]. Here let u = x2 + 3x - 1 with Dzu = 2x + 3 v =x4+7z+5with Dv =4x1+7. y = (x2+3x- 1)(4x1+7) + (x'+7x+5)(2x+3). Sec. 5 THE DERIVATIVE 54 The student should experience the extra trouble in multiplying the parenthetical expressions of the given function and then differentiating. 2 (B) y = x + 1' x X -1. [Form: y = u v DZy _ (x + 1) (2x) - x2 _ x(x + 2) (x + 1)2 (C) y = (x' + 1)1/2.
Form: y = u v DZy _ (x + 1) (2x) - x2 _ x(x + 2) (x + 1)2 (C) y = (x' + 1)1/2. (x + 1)2 [Form: y = u°] 3x2 D=y = '(x' + 1)-12(3x2) _ 2(x' + 1)1/2 ' (D) r = sin2O - cos 0. Here sine 0 is in the form u2, so that Der = 2( sin 9) cos B + sin B. 6. Differentiation of Implicit Functions The differentiation formulas just developed are especially useful in the differentiation of functions given implicitly. For example, consider x2+xy+y2-y- 1 =0. Here y represents values of two functions-those obtained by solving the quadratic for y in terms of x.
We list tables of values to give numerical evidence that f(x)->5. ->3 J ... -* 5 [The student should check these values for self-assurance and realize that no numerical array such as this can prove that f (x) -> 5]. Let us take a different approach to the problem for mathematical certainty by setting x=3+Ax where the symbol Ax (read "delta x") is a quantity that will be made arbitrarily near 0; thus, as Ox - 0, then x -i 3. Setting 3 + Ox in the function f( ) ( )2- ( ) -6 ( ) -3 we have f(3+Ax) _ (3 +x)2 - (3 + Ax) - 6 (3+Ax) -3 9+6(Ox) + (Ax)2- 3- (Ax) - 6 3+Ax-3 5(Ax) + (Ax)2 VARIABLES, FUNCTIONS, LIMITS, CONTINUITY 36 Sec.