Calculus: Analytic Geometry and Calculus, with Vectors

170 Functions, limits, derivatives

4 Each of the formulas

y = x2 sin x, y = xex

can be read "y equals u times v." Do this and obtain the derivatives

dy=x2 cosx+2xsinx,

dz =xex+ex.

5 Each of the formulas

Y =

sin x log x

x Y x

can be read "y equals u over v." Do this and obtain the formulas

dy x cos x - sin x dy_1 - log x

dx x2 ' dx x2

(^6) Derivatives of derivatives are called derivatives of higher order; the deriva-
tive off at x is f'(x), the derivative of f at x is f"(x), the derivative of f" atx is
f"(x) or fC3)(x), and so on. Supposing that % is a number and
f(x) =z
.+
x =(z + x)-1,
show that

f'(x) _ -(z + x)-2, f"(x) = 2!(z + x)-3, f(3)(x) = -3!(z + x)-4,

f(s)(x) = 4!(z + x)-5, f(l)(x) -5!(z + x)-S, f(s)(x) = 6!(z + x)-7,

where 2! = 1.2, 3! = 12.3, 4! = 1.2.3.4, etcetera. Supposing as usual that

0! = 1 and 1! = 1, observe that

f(%)(x)_ (-1)"n!(z + x)-a-1 (n = 0, 1, 2, ...)

when we agree that the result of differentiating f zero times is f itself.

(^7) Lettingf(x) = (1 - 2x)-1, show that
f(")(x)= 2^n!(1 - 2x)-"-1 (n = 0, 1, 2, ...).
(^8) Letting f(x) = log (1 + x2), show that
f, (x) 2x ,, 2 - 2x2
1 + x.2' J (x) _(1 + x2)2
and calculate one more derivative.
9 The formulas
(a) sin (a + b)x = sin ax cos bx + cosax sin bx
(b) cos (a + b)x = cos ax cos bx- sin ax sin bx
(c) e(a-Fb)x = eaxebx
(d) log ax = log a + log x
are permanently remembered by all good scientists. For each formula, calculate
the derivatives with respect to x of the two sides and show that the resultsare
equal.