Calculus: Analytic Geometry and Calculus, with Vectors

3.5 Difference quotients and derivatives 159

Problems 3.59

1 Give the definition of the derivative of f at x. 4nr.: (3.53) or (3.54).
2 Find dy/dx from the definition of derivatives and then check the answers
by use of formulas for differentiation when

(a) Y = %Ix

_ 1

(t) Y x+

(b)Y=N/X-

(d) Y 1 + x

(e)Y=4xx (f)Y=112x

1

(g) Y + x2 (h) Y 1x2

3 Sometimes we are given a formula for f(x) and are required to find the

derivative off at a, where a is a number given in decimal form. In some cases

it is easiest to find f'(a) directly from the formula

f,(a) = l. o

f(a + AX) - f(a)

a

or f'(a) = limf (a + h) - f(a)

A-0 h

In some cases it is easiest to work out a formula for f'(x) and put x = a in the

result. Work the problem both ways when

(a)f(x)=x2-3x+ 1, a = 416

(b) f (x) = x2 - 3x + 1, a = 0

(c)f(x)=1+x'a=2

(d) f(x) = 1 x x2, a = 0

J4 Formulas for derivativesare often wonderful, but there are times when it

is best to use the definition of derivatives to obtain f'(a). Letting g(x) = xIxj,

find g'(0) or show that g'(0) does not exist. Ans.:

g'(0) = limg(h) - g(0)= lim

hIhj 0.

a-+O It h-+o It

5 Supposing that

y = 1 +x+x2+_x3+.x4

z=5+4x-3x2+5x3+x4,

tell what facts or formulas or both enable us to write

dx =1 + 2x + 3x2 + 4x3

dz=4-6x+15x2+4x3.

dx