93 ■ Example 4.1.3 Consider the function h: R→Rgiven byh(x) = (3x 4 +x+7)15. Since h(x) is a polynomial we could in principle compute h′(x) by expanding the power and using Example 4.1.2. However, Theorem 4.1.3 provides a shorter way. Define f,g: R→Rby f (x) =3x4 +x+7 and g(x) =x15. Then h=g◦ f . Given x 0 ∈R, it follows from Theorem 4.1.3 that (g◦ f )′(x0) =g′(f (x0))f ′(x0) =15(3x 4 0 +x0 +7) 14(12x3 0 +1). ■ Example 4.1.4 By iterating the Chain Rule, we can extended the result to the composition of more than two functions in a straightforward way. For example, given functions f : I1 →R, g: I2 →R, andh: I3 →Rsuch that f (I1) ⊂I2, g(I2) ⊂I3, f is differentiable at x0, gis differentiable at f (x0), and h is differentiable at g(f (x0)), we obtain that h◦g◦ f is differentiable at x0 and (h◦g◦ f )′(x0) =h′(g(f (x0)))g′(f (x0))f ′(x0). Definition 4.1.2 Let I be an open interval in Rand let f : I →Rbe a differentiable function. If the function f ′ : I →Ris also differentiable, we say that f is twice differentiable (on I). The second derivative of f is denoted by f ′′ or f (2). Thus, f ′′ = (f ′)′. Similarly, we say that f is three times differentiable if f (2) is differentiable, and (f (2))′ is called the third derivative of f and is denoted by f ′′′ or f (3). We can define in this way n times differentiability and the nth derivative of f for any positive integer n. As a convention, f (0) =f . Definition 4.1.3 Let I be an open interval in Rand let f : I →R. The function f is said to be continuously differentiable onI if f is differentiable onI and f ′ is continuous onI. We denote by C1(I) the set of all continuously differentiable functions on I. If f is ntimes differentiable on I and the nth derivative is continuous, then f is called n times continuously differentiable on I. We denote byCn(I) the set of all ntimes continuously differentiable functions on I. Exercises 4.1.1 Prove parts (i) and (ii) of Theorem 4.1.2. 4.1.2 Compute the following derivatives directly from the definition. That is, do not use Theorem 4.1.2, but rather compute the appropriate limit directly (see Example 4.1.1). (a) f (x) =mx+bwhere m,b∈R. (b) f (x) = 1 x (here assume x̸ =0). (c) f (x) =√x (here assume x >0). 4.1.3 Let f : R→Rbe given by f (x) =( x2, if x >0; 0, if x ≤0. (a) Prove that f is differentiable at 0. Find f ′(x) for all x ∈R. (b) Is f ′ continuous? Is f ′ differentiable? 4.1.4 Let f (x) =( xα, if x >0; 0, if x ≤0.
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