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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 ...view middle of the document...

where u = g(x) and y = f (u) = f (g(x)).

INVERSE: LOGARITHMIC DIFF: FUNDAMENTAL TH.:

D(inv( f )) = 1 Dy g = 1/Dx f

(D f ◦ inv( f ))

where inv( f ) is the inverse function to f . where y = f (x) and x = g(y). but simplify ln | f | ﬁrst. if f is continuous at x.

D( f ) = f · D(ln | f |)

Z x

D

a

f dx = f (x)

6. Rules in special situations

On expressions like k · f (x) where k is constant do not use the product rule — use linearity. On expressions like 1/ f (x) do not use quotient rule — use the reciprocal rule, that is, rewrite this as f (x)−1 and use the Chain rule. Use logarithmic differentiation to avoid product and quotient rules on complicated products and quotients and also use it to differentiate powers that are messy. One can use b p = e p ln b to differentiate powers. Use logb |x| = ln |x|/ ln b to differentiate logs to other bases.

7. Rules for Elementary Functions

Dc = 0 D(ax + b) = a Dx p = px p−1 Dx = 1 Dx2 = 2x √ 1 D x= √ 2 x D1/x = −1/x2 D|x| = x/|x| = sgn(x) Dsgn(x) = 0 D x =D x =0 D sin x = cos x D cos x = − sin x D tan x = Dex = ex D ln x = 1/x D ln |x| = 1/x D arctan x = 1/(1 + x2 ) √ D arcsin x = 1/ 1 − x2 √ D arccos x = −1/ 1 − x2 Domain of derivative: x > 0. sec2 x = 1 + tan2 x D sec x = sec x tan x (derivatives of cotrigs always have minus signs.) where c is constant. where a and b are constant. p constant. For non-constant p use logarithmic diff. or rewrite as e p ln x . This is p = 1 in above. This is p = 2 in above. This is p = 1/2 in above. This is p = −1 in above. Domain of derivative: x = 0. Domain of derivative: x = 0. Domain of derivative: x not an integer (0, ±1, ±2, . . .). Note: |x| = √ x2

2

D cosh x = sinh x D sinh x = cosh x D tanh x = sech2 x = 1 − tanh2 x D sech x = − sech x tanh x D arctanh x = 1/(1 − x2 ) √ D arccosh x = 1/ −1 + x2 √ D arcsinh x = 1/ 1 + x2 Domain of...

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