35. Differentiation of a variable with respect to itself.
Following the General Rule, p. 29, we have
The derivative of a variable with respect to itself is unity.
36. Differentiation of a sum.
Let y = u -f- v w.
By the General Rule,
FIRST STEP. y + Ay = u + AM + v + Av w Aw.
SECOND STEP. A?/ = Aw + AV Aw.
Ay AM Av Aw
THIRD STEP. = 1
A# Ao: Aa: Aa:
c?y cZw c?v e?w
FOURTH STEP. -f- +
a3J dx dx dx
[Applying Th. I, p. 18.]
d , . du dv dw
III ' (" + v - w) = + 4- ~
dx dx dx dx
Similarly, for the algebraic sum of any finite number of functions.
The derivative of the algebraic sum of a finite number, of functions is equal to the same algebraic sum of their derivatives.
37. Differentiation of the product of a constant and a function.
Let y = cv.
By the General Rule,
FIRST STEP. y + Ay = c (v + Av) = cv + cAv.
SECOND STEP. Ay = c Av.
