Ex 3_5 - 3.5 The Chain Rute and Parametic Equations 773...

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Unformatted text preview: 3.5 The Chain Rute and Parametic Equations 773 Parametric Formnla for l'J/dr' If the equations 'I : l(t), l, = 3(, define I as a twice-diferentiable tunrion of r, then at any point where Al dt + 0 and f' = d, dx, d2y _ dy' /dl dxz dx/dt' EXAIiPLE 13 Fnd d7&'as a tunction of tifi" = t t2,J : t - t1. sottltlon l. Express l' : d/dr in terrns ofr dt dr, in Terft of r qrEs '' = /r/dr in tems of L-4'dt. trdr'dtbyd,/dt. , dr dy/dt 1 3t2 ' da .Lrldt | - 2t 2. Diflerentiate /' with respect to ,. dr' a(r-zt\ dt .lt \l 2t / 2 6t+6t2 (1 - 2t)z 3. Diyide dy'/dtby dxldt. dzy _,1r'ldt _(2 6t+6t,)1(J-2t)2 _2- 6t+6t2 dxz dx/dt | - 2t (l 2rl siven ,: l(,) a u = elx), fild dr/dr= Find the derivatives ofthe lunctions i! Exerciss 23+8 zl.p=t/t t 24.r=x?z+t)t1'z Eq. (3) h.is l {, !'lrr. !=6.r-9, u= (112\14 2.!=2u3, u:8n-1 ,r=sinz. u=3r+l 4.r,=cosr, ": -t3 _t = co5!. ,=sinr 6./:sinr, ,:x-cosl !=gdtu. u= 10r--5 8. y= -secu, u=l+tx hi!s 9 22, wile the ftnction in the fom ,l, : llr) and trr.Ir'en find d],/d{ as a tuncrion ofl :s....
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Ex 3_5 - 3.5 The Chain Rute and Parametic Equations 773...

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