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This method can be extended. If F ∈ C 2 (Ω), while the other assumptions are unchanged, then y = y(x) is also defined and of class C 2 for |x − x0 | < ε, where ε > 0 is chosen appropriately. Then y �� (x) is found by differentiating either (75) or (76) with respect to x. It is obvious how we proceed when F ∈ C n (Ω). 7 Differentiation of an inverse function. 6. Let f : I → J be one-to-one and strictly monotone and of class C 1 . There exists a unique inverse function f : J → I, which is also of class C 1 , such that (f ◦ g)(x) = f (g(x)) = x for x ∈ J.
They ought to be mastered by the students, so the differentiation formulæ in the applications should not be checked all the time. Power functions. If f (x) = xα , x ∈ R+ and α ∈ R, then (53) f � (x) = d (xα ) = α · xα−1 , dx for x ∈ R+ . √ We must only be careful with the case x = 0, because we are never allowed to divide by zero. g. 3 x is defined for all 1 x ∈ R, but its derivative x−2/3 is not. com 52 Differentiation Calculus 1a Logarithms. It follows from Chapter 2 that (54) 1 d ln x = x dx for x ∈ R+ , and d d logg (x) = (55) dx dx � ln x ln g � = 1 , x · ln g x ∈ R+ , where g ∈ R+ \ {1}.
Since the functions are of class C 1 with the derivative 0, they must be constants. g. x = 0. Hence, by a trivial continuous extension in the first case, Arccos x + Arcsin x = π 2 for x ∈ [−1, 1], π 2 for x ∈ R. 9 Magnitude of functions. In many cases we have to compare two functions, which both tend to infinity when x → +∞ or x → −∞. To ease matters in the following chapters some results are given here after our introduction of our most common functions in this chapter. Notice that even if we only consider four classes of functions, logarithms, power functions, exponentials and faculty functions, we get a lot of information from the fact that these four types of functions all tend to +∞ for x → +∞, but with a different speed.