Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/61

Rh '''35. Differentiation of a variable with respect to itself. '''

Let

Following the General Rule, p. 29, we have

Rh Rh Rh Rh II

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.