MATH1013 Tutorial 6 Two Special Limits eh − 1 = 1 since e is defined to be the number such that the limit equals 1. lim h→0 h That is, the only number such that y = ax has derivative 1 at x = 0. sin x = 1 by considering the following graph and comparing the areas of the circular sector lim x→0 x sin x and the two other triangles for positive x. The left hand limit is equal since is even. x 1 x 1 9h − 3h . h→0 h 1. Evaluate lim 2. Evaluate lim tan 3x . sin 8x 3. Evaluate lim 1 − cos x . x2 x→0 x→0 Derivatives of Trigonometric Functions The derivatives of the six trigonometric functions are summarized as follows: d sin x = cos x dx d cos x = − sin x dx d tan x = sec2 x dx d cot x = − csc2 x dx d sec x = sec x tan x dx d csc x = − csc x cot x dx We can obtain the derivatives of the co-functions by adding negative signs and switching each corresponding term from the derivative of the three trigonometric functions. Prepared by Leung Ho Ming Homepage: http://ihome.ust.hk/~malhm 1 The Chain Rule If g is differentiable at x and f is differentiable at g(x), then the composite function F = f ◦ g is differentiable at x and F 0 (x) = f 0 (g(x)) · g 0 (x). In Leibniz notation, if y = f (u) and g = u(x) are differentiable functions then dy du dy = · . dx du dx 4. Find the derivatives of the following functions. (a) f (x) = (x2 + sec x + 1)3 (b) g(x) = sin x2 cos 2x (c) h(x) = 4 sin (d) p(x) = p 1+ √ x cos x 1 + sin x 5. Suppose that the functions f and g and their derivatives have the following values at x = 0 and x = 1 x 0 1 f (x) 1 3 g(x) 1 −4 f 0 (x) 5 −1/3 g 0 (x) 1/3 −8/3 Find the derivatives with respect to x of the following combinations at x = 0 (a) f (x)(g(x))3 (b) f (g(x)) (c) g(f (x)) (d) f (x + g(x)) 2 Implicit Differentiation Suppose the relation between x and y is given as f (x, y) = 0, for example x2 + y 2 − r2 = 0. To obtain the derivative dy/dx, we simply differentiate both sides by treating y as a function of x. With implicit differentiation, we can obtain the derivatives of the six inverse trigonometric functions: d 1 sin−1 x = √ dx 1 − x2 d 1 cos−1 x = − √ dx 1 − x2 d 1 tan−1 x = dx 1 + x2 d 1 cot−1 x = − dx 1 + x2 d 1 √ sec−1 x = dx |x| x2 − 1 d 1 csc−1 x = − √ dx |x| x2 − 1 Indeed only the derivatives of sin−1 x and tan−1 x are important, and are suggested to be memorized. 6. Show that d 1 √ sec−1 x = for |x| > 1 by implicitly differenting y = sec−1 x, or sec y = x. dx |x| x2 − 1 x2 y2 7. An ellipse centered at origin that passes through (−a, 0), (a, 0), (0, −b) and (0, b) has equation 2 + 2 = 1 where a b x0 x y0 y a, b > 0. Show that the equation of tangent line at (x0 , y0 ) is 2 + 2 = 1. a b 8. Suppose f is a one-to-one differentiable function and its inverse f −1 is also differentiable. (a) Differentiate f ◦ f −1 (x) = x implicitly to show that (f −1 )0 (x) = (b) If f (4) = 5 and f 0 (4) = 32 , find (f −1 )0 (5). 3 1 . f 0 (f −1 (x)) Derivatives of Exponential and Logarithmic Functions The derivative of ex is ex . If the exponent is not in base e, then d x d x ln a a = e = ex ln a · ln a = ax ln a. dx dx d 1 ln x = . If the logarithm is not in base e, then dx x The derivative of ln x is d ln x 1 d loga x = = . dx dx ln a x ln a If f (x) is a complicted function involving products, quotients or powers, we may apply logarithms: y = f (x) ⇒ ln y = ln f (x) and differentiate implicitly to obtain f 0 (x). 9. Find the derivatives of the functions. 5 (x + 1)(x − 1) where x > 2. (a) f (x) = (x − 2)(x + 3) (b) g(x) = (x + 2)x+2 √ (c) h(x) = (sin x) x 4