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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 y being some function of X, and u some function of y. This theorem may be extended to any number of functions,

dy'da

so that

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Ex. (1)
Let U =

(a + bar")".
Then y= a + bx", 4 = "",
dy

du nbach,

= my-) = m (a + ")"-'; dr

dy

du
therefore mnbon-'(a + bw*)-1.
da

du {x + (1 + x)}}
(2)
{x + (1 + )}},

da 2 (1 + 2*)

du (3)

nan-16 dc

3 (4)

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1

[ocr errors]

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

x log x (7) u = log” x; which signifies not the nth power of the logarithm of x, but the nth logarithm of that quantity,

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