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If x=cosθ-rsinθ, y=sinθ+rcosθ prove that dr/dx = x/r
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x=cosθrsinθ, y=sinθ+rcosθ Squaring and adding, x2+y2=(cosθrsinθ)2+(sinθ+rcosθ)2 x2+y2=cos2θ2rsinθcosθ+r2sin2θ+sin2θ+2rsinθcosθ+r2cos2 x2+y2=(cos2θ+sin2θ)+r2(cos2θ+sin2θ)=1+r2 Implicitly differentiating x2+y2=1+r2, w.r.t x, 2x+0=0+2rdrdx drdx=xr

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