# D dt 5 sin 2 t g s t d d dt tan 5 sin 2 t 5 sin 2 t g

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Chapter 4 / Exercise 4
Intermediate Algebra: Connecting Concepts through Applications
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d dt (5 - sin 2 t ) g / s t d = d dt (tan (5 - sin 2 t )) 5 - sin 2 t , g s t d = tan s 5 - sin 2 t d . Derivative of tan u with u = 5 - sin 2 t Derivative of with u = 2 t 5 - sin u H ISTORICAL B IOGRAPHY Johann Bernoulli (1667–1748) Power Chain Rule with u = 5 x 3 - x 4 , n = 7 Power Chain Rule with u = 3 x - 2, n = - 1 because sin n x means s sin x d n , n , - 1. Chapter 2: Differentiation 109 02 C2 Differentiation.indd 109 7/27/12 1:19:24 PM
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Chapter 4 / Exercise 4
Intermediate Algebra: Connecting Concepts through Applications
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EXAMPLE 6 In Section 2 .2, we saw that the absolute value function is not differentiable at x # 0. However, the function is differentiable at all other real numbers as we now show. Since , we can derive the following formula: EXAMPLE 7 Show that the slope of every line tangent to the curve is positive. Solution We find the derivative: Power Chain Rule with At any point ( x , y ) on the curve, and the slope of the tangent line is the quotient of two positive numbers. EXAMPLE 8 The formulas for the derivatives of both sin x and cos x were obtained un- der the assumption that x is measured in radians, not degrees. The Chain Rule gives us new insight into the difference between the two. Since radians, radi- ans where x ° is the size of the angle measured in degrees. By the Chain Rule, See Figure 2 .25. Similarly, the derivative of The factor would compound with repeated differentiation. We see here the advantage for the use of radian measure in computations. p > 180 cos s x ° d is - s p > 180 d sin s x ° d . d dx sin s x ° d = d dx sin ( p x 180 ) = p 180 cos ( p x 180 ) = p 180 cos s x ° d . x ° = p x > 180 180° = p dy dx = 6 s 1 - 2 x d 4 , x , 1 > 2 = 6 s 1 - 2 x d 4 = - 3 s 1 - 2 x d - 4 ! s - 2 d u = s 1 - 2 x d , n = - 3 = - 3 s 1 - 2 x d - 4 ! d dx s 1 - 2 x d dy dx = d dx s 1 - 2 x d - 3 y = 1 > s 1 - 2 x d 3 = x 0 x 0 , x , 0. = 1 2 0 x 0 ! 2 x = 1 2 2 x 2 ! d dx ( x 2 ) d dx ( 0 x 0 ) = d dx 2 x 2 0 x 0 = 2 x 2 y = 0 x 0 Power Chain Rule with u = x 2 , n = 1 > 2, x , 0 2 x 2 = 0 x 0 Derivative of the Absolute Value Function d dx ( 0 x 0 ) = x 0 x 0 , x , 0 x y 1 180 y & sin x y & sin( x \$ ) & sin x 180 FIGURE 2 .25 oscillates only times as often as oscillates. Its maximum slope is at (Example 8). x = 0 p > 180 sin x p > 180 Sin s x ° d 110 Part 1: Calculus 02 C2 Differentiation.indd 110 7/27/12 1:19:33 PM
made of metal, its length will vary with temperature, either in- creasing or decreasing at a rate that is roughly proportional to L . In symbols, with u being temperature and k the proportionality constant, Assuming this to be the case, show that the rate at which the pe- riod changes with respect to temperature is .
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