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DIFFERENTIAL CALCULUS.

CHAPTER I.

DIFFERENTIATION.

Functions of One Variable.

If u be an explicit function of X, which is of a complicated form, it may generally be reduced to the differentiation of simpler functions by means of the theorem

du du dy

dx dy 'dx y being some function of x, and u some function of y. This theorem may be extended to any number of functions,

so that

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dy

Ex. (1) Let U =

(a + 600*)". Then y= a + b2",

y",
dy

du
= nb.-!, = my-) = m (a + b2")");
dr

du
therefore = mnban-'(a + b.w*)"-1.
dc

du {x + (1 + x2)"}! (2) = {x + (1 + 0*)},

2 (1 + 0*)!
du
U = eth; = nxn-en

dc

da

(3)

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1

(5)

u = log {x + (1 + 2;?)}}

du d x

(1 + c^)

1

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du (6) log (log x) = log2 (x);

dir x log x (7)

log" v; which signifies not the nth power of the logarithm of x, but the nth logarithm of that quantity,

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