EXAMPLE 6In Section 2.2, we saw that the absolute value function is notdifferentiable at x#0. However, the function isdifferentiable at all other real numbers aswe now show. Since , we can derive the following formula:EXAMPLE 7Show that the slope of every line tangent to the curve ispositive.SolutionWe find the derivative:Power Chain Rule with At any point (x,y) on the curve, and the slope of the tangent line isthe quotient of two positive numbers.EXAMPLE 8The formulas for the derivatives of both sin xand cos xwere obtained un-der the assumption that xis measured in radians, notdegrees. The Chain Rule gives us newinsight 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>180cossx°dis -sp>180dsinsx°d.ddxsinsx°d=ddxsin(px180)=p180cos(px180)=p180cossx°d.x°=px>180180°=pdydx=6s1-2xd4,x,1>2=6s1-2xd4= -3s1-2xd-4!s-2du=s1-2xd,n= -3= -3s1-2xd-4!ddxs1-2xddydx=ddxs1-2xd-3y=1>s1-2xd3=x0x0,x,0.=120x0!2x=122x2!ddx(x2)ddx(0x0)=ddx2x20x0=2x2y=0x0Power Chain Rule withu=x2,n=1>2,x,02x2=0x0Derivative of the Absolute Value Functionddx(0x0)=x0x0,x,0xy1180y&sin xy&sin(x$)&sin�x180FIGURE2.25oscillates only times as often as oscillates. Itsmaximum slope is at (Example 8).x=0p>180sinxp>180Sinsx°d110Part 1:Calculus02 C2 Differentiation.indd 1107/27/12 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 ubeing temperature and kthe proportionalityconstant,Assuming this to be the case, show that the rate at which the pe-riod changes with respect to temperature is .