Question

real analysis

1,2,3,4,8please

5.1.5a

Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative of tan xsin x/cos x. 4. Prove Theorem 5.1.5(a). Spre o e fong eermin eter the give function is differentiable at the indicated point t hau your answer! e, f(x) = { sin x, x E Q. Let f)x. Compute )/"(), and show thatf"(0) does not exist. Determine where each of the following function a. f)x] from R to R is differentiable and find the derivative. *t, h(x)= /sin xl 8. Use the product rule, quotient rule, and chain rule to find the derivative of each of the following functions a, f(x) = x sing, x*0 c, f(x) = Vx + V2 + x b, f(x) = (cos (sin x)")", n, m E N d. f(x) = x"(2 + sin), x #0 Since f(r)-f(P) lim exists and equals p' (p), by Theorem 4,1.6) Therefore, limo)p) and thus fis continuous at p. In the above, if p is an endpoint. then the limits are either the right or left limit at p, whichever is appropriate. Remark. In both Examples 5,1.3(d) and (F), the given function is continuous at 0, bu not differentiable at 0. Given a finite number of points, say P. P it is easy to con struct a functionf which is continuous but not differentiable at pl··.pr For examp has the desired properties. In 1861, Weierstrass constructed a function f which is tinuous at every point of R but nowhere differentiable. When published in 1874 example astounded the mathematical community. Prior to this time mathemati generally believed that continuous functions were differentiable (except perhap finite number of points). In Example 8.5.3 we will consider the function of Weie in detail. Derivatives of Sums, Products, and Quotients We now derive the formulas for the derivative of sums, products, and quotient tions. These rules were discovered by Leibniz in 1675. Suppose f, g are real-valued functions defined on an interval THEOREM 5.1.5 g are differentiable at x E I, then f + 8.fs, and fig (if 8(x) + 0) are ditfe x and )fuetev (8(x)2 (c)()(x) = f (x)g(x)-))(x)g (s), provi dede( Proof. The proof of (a) is left as an exercise (Exercise adding and subtracting the term f(x + h)g(x), we have f Beh-(l)x th x + h)

Answer 1

036 From -92