Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 72. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Differentiating sin x and cos x 691 Proof that lim . .0 cos h 1 0 But cos h - 1 lim lim lim lim lim lim cos h 1 cos h 1 h cos h 1 cos2 h 1 1 h cos h 1 1 cos2 h 1 h cos h 1 sin2 h 1 h co is h 1 sin h sin h cos h 1 sin h lim ------ h sin h and lim cos h 1 1 1 0 0 h 1 h so cos h 1 lim ---- ------ 1 0 0. h We have now shown that sin x cos x as conjectured The Chain Rule tells us that sin g x cos g x g x . Now that we have proven X sin x cos x the derivative of cos x is easy to tackle by using the fact that cos x and sin x are related to one another by a horizontal shift. Looking back at the graphs of cos x and its derivative it is easy to speculate that the derivative of cos x is - sin x . Proof that the Derivative of cos x is sin x Observe that sin x y cos x replacing x by x y shifts the sine graph left y units. Similarly cos x 2 sin x replacing x by x 2 shifts the cosine graph left 2 units. sin x f x cos x g x Figure 692 CHAPTER 21 Differentiation of Trigonometric Functions cos x sin x dx dx 2 cos by the Chain Rule sin x Combining this result with the Chain Rule gives us sin g x cos g x g x or informally dx sin mess cosUmess mess dx cos g x sin g x g x dx cos mess sin mess mess dx 1 2 where mess is a function of x. Using the Chain Rule and either the Product or Quotient Rule allows us to flnd the derivatives of tan x csc x sec x and cot x. EXERCISE EXERCISE EXAMPLE SOLUTIONS Differentiate the following. a y 3x sin x2 b y 7 cos2 3x 5 c y tan x2 a This is the product of 3x and sin ess . y 3 sin x2 3x cos x2 2x 3 sin x2 6x2 cos x2 b 7 cos2 3x 5 7 cos 3x 5 2 so basically this is 7 ess 2 and its derivative is 14 ess ess where the ess is cos 3x 5 . Then the Chain Rule must be applied to cos 3x 5 . y 14 cos 3x 5 - sin 3x 5 3 -42 cos 3x 5 sin 3x 5 c y tan x2 tan x2 1 2. This is basically ess 1 2 so its derivative is 2 ess -1 2 ess . We know ess tan stuff so ess sec2 stuff stuff . y 1 2 tan x2 j sec2 x2 2x x sec2 x2
