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y f 0 x means that for any " > 0 On the other hand, dy and y are related. In particular, lim x!0 x y dy < " whenever j xj < . Now dx is an independent variable there exists  > 0 such that " < x dx and the axes of x and dx are parallel; therefore, dx may be chosen equal to x. With this choice " x < y dy < " x or dy " x < y < dy " x From this relation we see that dy is an approximation to y in small neighborhoods of x. dy is called the principal part of y. dy The representation of f 0 by has an algebraic suggestiveness that is very appealing and will appear dx in much of what follows. In fact, this notation was introduced by Leibniz (without the justi cation provided by knowledge of the limit idea) and was the primary reason his approach to the calculus, rather than Newton s was followed.
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THE DIFFERENTIATION OF COMPOSITE FUNCTIONS Many functions are a composition of simpler ones. For example, if f and g have the rules of correspondence u x3 and y sin u, respectively, then y sin x3 is the rule for a composite function F g f . The domain of F is that subset of the domain of F whose corresponding range values are in the domain of g. The rule of composite function di erentiation is called the chain rule and is represented dy dy du F 0 x g 0 u f 0 x . by dx du dx In the example dy d sin x3  cos x3 3x2 dx dx dx The importance of the chain rule cannot be too greatly stressed. Its proper application is essential in the di erentiation of functions, and it plays a fundamental role in changing the variable of integration, as well as in changing variables in mathematical models involving di erential equations.
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IMPLICIT DIFFERENTIATION The rule of correspondence for a function may not be explicit. For example, the rule y f x is implicit to the equation x2 4xy5 7xy 8 0. Furthermore, there is no reason to believe that this equation can be solved for y in terms of x. However, assuming a common domain (described by the independent variable x) the left-hand member of the equation can be construed as a composition of functions and di erentiated accordingly. (The rules of di erentiation are listed below for your review.) In this example, di erentiation with respect to x yields     dy dy 2x 4 y5 5xy4 7 y x 0 dx dx Observe that this equation can be solved for dy as a function of x and y (but not of x alone). dx
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RULES FOR DIFFERENTIATION If f , g; and h are di erentiable functions, the following di erentiation rules are valid. 1: d d d f f x g x g f x g x f 0 x g 0 x dx dx dx d d d f f x g x g f x g x f 0 x g 0 x dx dx dx d d fC f x g C f x C f 0 x where C is any constant dx dx d d d f f x g x g f x g x g x f x f x g 0 x g x f 0 x dx dx dx & ' g x d f x f x d g x d f x g x f 0 x f x g 0 x dx dx 2 dx g x g x g x 2 (Addition Rule)
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