Differentiation Rules (Differential Calculus)

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Differentiation Rules (Differential Calculus)
1. Notation
The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x.

2. Alternate Notations for (D f )(x) f (x) d For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), d dx , d f (x), f (x), f (1) (x). The dx “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy whereas f (x) is the value of it at x. If y = f (x), then Dx y, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f , “∆ f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy Historical note: Newton used y, while Leibniz used dx . About a century later Lagrange introduced y and ˙ Arbogast introduced the operator notation D.

3. Domains
The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below.

4. Operating Principle
Many functions are formed by successively combining simple functions, using constructions such as sum, product and composition. To differentiate, apply the differentiation rule corresponding to the last construction.

5. Rules for Constructions
SUM: LINEARITY: PRODUCT: RECIPROCAL: QUOTIENT: CHAIN (for compositions): D( f + g) = D f + Dg D(a f + bg) = aD f + bDg D( f · g) = D f · g + f · Dg D(1/ f ) = −D f / f 2 g · D f − f · Dg g2 D( f ◦ g) = (D f ) ◦ g · Dg dy dy du = dx du dx D( f /g) = 1…...

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