Calculus: Analytic Geometry and Calculus, with Vectors

154 Functions, limits, derivatives

the manner in which it is applied can be (or should be) remembered with
the aid of the famous "four-step rule." We may not always get 4 when
we count the steps, but the rule is the four-step rule anyway.

Four-step rule 3.55

Definition A4pplication

Y = f (X) Y = x2

Y+Dy =f(x+Ox) y+Ay = (x+11x)2 = x2+2xAx+Ax2

Ay = Ax + Ax) - AX) Ay = 2x six + zx2

Ay=Ax + Ox) - AX)

'&y = 2x + Ox

Ox Ox 9x

Ay

dxI o Ax

4

dx =

2x

The steps are as follows: select (or "fix") an x in the domain of f, write

y = f(x), introduce Ox, write y + Ay = f(x + Ax), subtract to get Ay,

divide by Ax to get Ay/Ax, and, finally, find the limit as Ax --+ 0 to obtain

dy/dx. Whether we consciously use the four-step rule or not, we all

need experience in the art of calculating derivatives by finding limits of

difference quotients, and problems at the end of this section provide

some of it. Meanwhile, we gain experience by proving the following

formulas which can be and must be remembered.

Theorem 3.56 If u and v are differentiable functions of x and if c

and n are constants, then

(3.561)

(3.562)

(3.563)

(3.564)

(3.565)

(3.566)

d(u +v)

=dx+dx

d _ du

dx cu - c dx

d xrtnx"_I

d u" = nun-Idu

dx dx

dxuv - udxvdx

du dv

d_uadx- udxTX-Vu

x v v2

provided that v s 0 in (3.566) and that in (3.563) and (3.564) we have

x 0 0 and is s 0 when n is a negative integer and (except in some special

cases) x > 0 and u > 0 when n is not an integer.

The first three of these formulas enable us to obtain results like

Z(x4-3x'+5x2-7x+6) =4x$ -9x2+1Ox- 7