Calculus: Analytic Geometry and Calculus, with Vectors

3.6 The chain rule and differentiation of elementary functions 173

we can differentiate with respect to x with the aid of the chain rule to obtain

(2) 3y2dxmdx=1 or 3[y(x)]2y'(x) + y'(x) = 1

and hence

dy (^11)
(3) or y,(x)=
dx 3y2 + 1 3fy(x)]2 + 1.
From (3) and the assumption that y is a differentiable function of x, we see that
the derivative itself is a differentiable function of x. The derivative of the
derivative is called the second derivative and is denoted by the symbols in the
left members of the formulas
(4) d2y
-6y -6y
dx2 - (3y2 + 1)3' Y "(X) = (3y2 + 1)1
By differentiating the members of (3), show that the formulas (4) are correct.
(^20) Supposing that y is a differentiable function of x for which the given rela
tion holds, differentiate with respect to x to find dy/dx when
(a) xy = 7
(b) sin y = x
(c) ey = x
(d) sin xy = x + y
21 Find f'(x) when
(a) f(x) = log (x + .\/a2 + x2)
(b) f (x) = log ( a2 + x2 - x)
(c) f(x) = (log sin 2x)2
(d) f(x) = log (sin 2x)2
(e) f(x) = log sin (2x)2
n

(f) f(x) = (1 +C)

flns.:

dy

y

flnr.:dx cos yor

± 1 x

e4ns.: dx = ey or x

flns.: dx 1 - x cos xy

.dns a2+x2^1

-1

Ans.:

a2+ x2

Ans.:4 cos 2x log sin 2,x

sin 2x

fins :4 cos 2x

sin 2x

flns.:8x cos 4x2

sin 4x2

Ins.:n[1 + (1 - 2x)e2z]xn-l

(1 + e2x)n+l

22 The Hermite polynomials, depending upon a parameter a that is usually
taken to be 1 or 2 in applications, are defined by the formulas Ho(x) = 1 and

n

(1) HH(x) = (-1)neax2/2 __ a axe/2dXn (n > 1).