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Sec. 6] Differentials of First and Higher Orders 71
Sec. 6. Differentials of First and Higher Orders
1. First-order differential. The differential (first-order) of a function
y = f(x) is the principal part of its increment, which part is linear relative
to the increment Ax = dx of the independent variable x. The differential of a
Fig. 19
function is equal to the product of its derivative by the differential of the
independent variable
whence
dy--=y'dx,
, dy
uy
dx '
If MN is an arc of the graph of the function y = f(x) (Fig. 19), MT is the
tangent at M. (x, y) and
PQ = Ax-=dx,
then the increment in the ordinate of the tangent
and the segment AN
Example 1. Find
y = 3x
by.
the increment and the differential of the function
2
x.
Solution. First method:
or
Hence,
Second method:
A// = 3 (x + Ax) 2
(x + Ax)
At/ = (6* 1) Ax + 3 (Ax) 2 .
dy = (6x 1) Ax = (6x 1) dx.
3x 2 + x
t/' = 6x 1; df/ = j/'dx = (6x 1) dx.
Example 2. Calculate At/ and dy of the function y = 3x 2
and Ax = 0.01.
and
Solution. A/ = (6x l)-Ax + 3 (Ax) 2 = 5- 0.0 1 + 3- 2
(0.01 = ) 0.0503