Find dy/dx by implicit differentiation. 14. Y sin(x^2) = x sin(y^2) 18. Tan(x-y) = y/1+x^2
A asimon2005 New member Joined Nov 29, 2007 Messages 34 Oct 5, 2009 #1 Find dy/dx by implicit differentiation. 14. Y sin(x^2) = x sin(y^2) 18. Tan(x-y) = y/1+x^2
S soroban Elite Member Joined Jan 28, 2005 Messages 5,586 Oct 5, 2009 #2 Hello, asimon2005! Find dydx by implicit differentiation..\displaystyle \text{Find }\frac{dy}{dx}\text{ by implicit differentiation..}Find dxdy by implicit differentiation.. . . 14. y⋅sin(x2) = x⋅sin(y2)\displaystyle 14.\;\;y\cdot\sin(x^2) \:=\: x\cdot\sin(y^2)14.y⋅sin(x2)=x⋅sin(y2) Click to expand... Product Rule: y ⋅ cos(x2) ⋅ 2x+dydx ⋅ sin(x2) = x ⋅ cos(y2) ⋅ 2y ⋅ dydx+1 ⋅ sin(y2)\displaystyle \text{Product Rule: }\;y\!\cdot\!\cos(x^2)\!\cdot\!2x + \frac{dy}{dx}\!\cdot\!\sin(x^2) \;=\;x\!\cdot\!\cos(y^2)\!\cdot\!2y\!\cdot\!\frac{dy}{dx} + 1\!\cdot\!\sin(y^2)Product Rule: y⋅cos(x2)⋅2x+dxdy⋅sin(x2)=x⋅cos(y2)⋅2y⋅dxdy+1⋅sin(y2) We have: 2xycos(x2)+sin(x2)dydx = 2xycos(y2)dydx+sin(y2)\displaystyle \text{We have: }\:2xy\cos(x^2) + \sin(x^2)\frac{dy}{dx} \;=\; 2xy\cos(y^2)\frac{dy}{dx} + \sin(y^2)We have: 2xycos(x2)+sin(x2)dxdy=2xycos(y2)dxdy+sin(y2) Rearrange terms: sin(x2)dydx−2xycos(y2)dydx = sin(y2)−2xycos(x2)\displaystyle \text{Rearrange terms: }\;\sin(x^2)\frac{dy}{dx} - 2xy\cos(y^2)\frac{dy}{dx} \;=\;\sin(y^2) - 2xy\cos(x^2)Rearrange terms: sin(x2)dxdy−2xycos(y2)dxdy=sin(y2)−2xycos(x2) Factor: [sin(x2)−2xycos(y2)]dydx = sin(y2)−2xycos(x2)\displaystyle \text{Factor: }\;\bigg[\sin(x^2) - 2xy\cos(y^2)\bigg]\frac{dy}{dx} \;=\;\sin(y^2) - 2xy\cos(x^2)Factor: [sin(x2)−2xycos(y2)]dxdy=sin(y2)−2xycos(x2) Therefore: dydx = sin(y2)−2xycos(x2)sin(x2)−2xycos(y2)\displaystyle \text{Therefore: }\;\frac{dy}{dx} \;=\;\frac{\sin(y^2) - 2xy\cos(x^2)}{\sin(x^2) - 2xy\cos(y^2)}Therefore: dxdy=sin(x2)−2xycos(y2)sin(y2)−2xycos(x2) 18. tan(x−y) = y1+x2\displaystyle 18.\;\;\tan(x-y) \:=\:\frac{y}{1+x^2}18.tan(x−y)=1+x2y Click to expand... We have: (1+x2)⋅tan(x−y) = y\displaystyle \text{We have: }\;(1+x^2)\cdot\tan(x-y) \;=\;yWe have: (1+x2)⋅tan(x−y)=y Product Rule: (1+x2) ⋅ sec2(x−y) ⋅ (1−dydx)+(2x) ⋅ tan(x−y) = dydx\displaystyle \text{Product Rule: }\;(1 + x^2)\!\cdot\!\sec^2(x-y)\!\cdot\!\left(1 - \frac{dy}{dx}\right) + (2x)\!\cdot\!\tan(x-y) \;=\;\frac{dy}{dx}Product Rule: (1+x2)⋅sec2(x−y)⋅(1−dxdy)+(2x)⋅tan(x−y)=dxdy We have: (1+x2) ⋅ sec2(x−y)−(1+x2) ⋅ sec2(x−y)dydx+2x ⋅ tan(x−y) = dydx\displaystyle \text{We have: }\;(1+x^2)\!\cdot\!\sec^2(x-y) - (1+x^2)\!\cdot\!\sec^2(x-y)\frac{dy}{dx} + 2x\!\cdot\!\tan(x-y) \;=\; \frac{dy}{dx}We have: (1+x2)⋅sec2(x−y)−(1+x2)⋅sec2(x−y)dxdy+2x⋅tan(x−y)=dxdy Rearrange terms: dydx + (1+x2)sec2(x−y)dydx = (1+x2)sec2(x−y) + 2xtan(x−y)\displaystyle \text{Rearrange terms: }\;\frac{dy}{dx} \;+\; (1+x^2)\sec^2(x-y)\frac{dy}{dx} \;=\;(1+x^2)\sec^2(x-y) \;+\; 2x\tan(x-y)Rearrange terms: dxdy+(1+x2)sec2(x−y)dxdy=(1+x2)sec2(x−y)+2xtan(x−y) Factor: [1+(1+x2)sec2(x−y)]dydx = (1+x2)sec2(x−y)+2xytan(x−y)\displaystyle \text{Factor: }\;\bigg[1 + (1+x^2)\sec^2(x-y)\bigg]\frac{dy}{dx} \;=\;(1+x^2)\sec^2(x-y) + 2xy\tan(x-y)Factor: [1+(1+x2)sec2(x−y)]dxdy=(1+x2)sec2(x−y)+2xytan(x−y) Therefore: dydx = (1+x2)sec2(x−y)+2xytan(x−y)1+(1+x2)sec2(x−y)\displaystyle \text{Therefore: }\;\frac{dy}{dx} \;=\;\frac{(1+x^2)\sec^2(x-y) + 2xy\tan(x-y)}{1 + (1+x^2)\sec^2(x-y)}Therefore: dxdy=1+(1+x2)sec2(x−y)(1+x2)sec2(x−y)+2xytan(x−y)
Hello, asimon2005! Find dydx by implicit differentiation..\displaystyle \text{Find }\frac{dy}{dx}\text{ by implicit differentiation..}Find dxdy by implicit differentiation.. . . 14. y⋅sin(x2) = x⋅sin(y2)\displaystyle 14.\;\;y\cdot\sin(x^2) \:=\: x\cdot\sin(y^2)14.y⋅sin(x2)=x⋅sin(y2) Click to expand... Product Rule: y ⋅ cos(x2) ⋅ 2x+dydx ⋅ sin(x2) = x ⋅ cos(y2) ⋅ 2y ⋅ dydx+1 ⋅ sin(y2)\displaystyle \text{Product Rule: }\;y\!\cdot\!\cos(x^2)\!\cdot\!2x + \frac{dy}{dx}\!\cdot\!\sin(x^2) \;=\;x\!\cdot\!\cos(y^2)\!\cdot\!2y\!\cdot\!\frac{dy}{dx} + 1\!\cdot\!\sin(y^2)Product Rule: y⋅cos(x2)⋅2x+dxdy⋅sin(x2)=x⋅cos(y2)⋅2y⋅dxdy+1⋅sin(y2) We have: 2xycos(x2)+sin(x2)dydx = 2xycos(y2)dydx+sin(y2)\displaystyle \text{We have: }\:2xy\cos(x^2) + \sin(x^2)\frac{dy}{dx} \;=\; 2xy\cos(y^2)\frac{dy}{dx} + \sin(y^2)We have: 2xycos(x2)+sin(x2)dxdy=2xycos(y2)dxdy+sin(y2) Rearrange terms: sin(x2)dydx−2xycos(y2)dydx = sin(y2)−2xycos(x2)\displaystyle \text{Rearrange terms: }\;\sin(x^2)\frac{dy}{dx} - 2xy\cos(y^2)\frac{dy}{dx} \;=\;\sin(y^2) - 2xy\cos(x^2)Rearrange terms: sin(x2)dxdy−2xycos(y2)dxdy=sin(y2)−2xycos(x2) Factor: [sin(x2)−2xycos(y2)]dydx = sin(y2)−2xycos(x2)\displaystyle \text{Factor: }\;\bigg[\sin(x^2) - 2xy\cos(y^2)\bigg]\frac{dy}{dx} \;=\;\sin(y^2) - 2xy\cos(x^2)Factor: [sin(x2)−2xycos(y2)]dxdy=sin(y2)−2xycos(x2) Therefore: dydx = sin(y2)−2xycos(x2)sin(x2)−2xycos(y2)\displaystyle \text{Therefore: }\;\frac{dy}{dx} \;=\;\frac{\sin(y^2) - 2xy\cos(x^2)}{\sin(x^2) - 2xy\cos(y^2)}Therefore: dxdy=sin(x2)−2xycos(y2)sin(y2)−2xycos(x2) 18. tan(x−y) = y1+x2\displaystyle 18.\;\;\tan(x-y) \:=\:\frac{y}{1+x^2}18.tan(x−y)=1+x2y Click to expand... We have: (1+x2)⋅tan(x−y) = y\displaystyle \text{We have: }\;(1+x^2)\cdot\tan(x-y) \;=\;yWe have: (1+x2)⋅tan(x−y)=y Product Rule: (1+x2) ⋅ sec2(x−y) ⋅ (1−dydx)+(2x) ⋅ tan(x−y) = dydx\displaystyle \text{Product Rule: }\;(1 + x^2)\!\cdot\!\sec^2(x-y)\!\cdot\!\left(1 - \frac{dy}{dx}\right) + (2x)\!\cdot\!\tan(x-y) \;=\;\frac{dy}{dx}Product Rule: (1+x2)⋅sec2(x−y)⋅(1−dxdy)+(2x)⋅tan(x−y)=dxdy We have: (1+x2) ⋅ sec2(x−y)−(1+x2) ⋅ sec2(x−y)dydx+2x ⋅ tan(x−y) = dydx\displaystyle \text{We have: }\;(1+x^2)\!\cdot\!\sec^2(x-y) - (1+x^2)\!\cdot\!\sec^2(x-y)\frac{dy}{dx} + 2x\!\cdot\!\tan(x-y) \;=\; \frac{dy}{dx}We have: (1+x2)⋅sec2(x−y)−(1+x2)⋅sec2(x−y)dxdy+2x⋅tan(x−y)=dxdy Rearrange terms: dydx + (1+x2)sec2(x−y)dydx = (1+x2)sec2(x−y) + 2xtan(x−y)\displaystyle \text{Rearrange terms: }\;\frac{dy}{dx} \;+\; (1+x^2)\sec^2(x-y)\frac{dy}{dx} \;=\;(1+x^2)\sec^2(x-y) \;+\; 2x\tan(x-y)Rearrange terms: dxdy+(1+x2)sec2(x−y)dxdy=(1+x2)sec2(x−y)+2xtan(x−y) Factor: [1+(1+x2)sec2(x−y)]dydx = (1+x2)sec2(x−y)+2xytan(x−y)\displaystyle \text{Factor: }\;\bigg[1 + (1+x^2)\sec^2(x-y)\bigg]\frac{dy}{dx} \;=\;(1+x^2)\sec^2(x-y) + 2xy\tan(x-y)Factor: [1+(1+x2)sec2(x−y)]dxdy=(1+x2)sec2(x−y)+2xytan(x−y) Therefore: dydx = (1+x2)sec2(x−y)+2xytan(x−y)1+(1+x2)sec2(x−y)\displaystyle \text{Therefore: }\;\frac{dy}{dx} \;=\;\frac{(1+x^2)\sec^2(x-y) + 2xy\tan(x-y)}{1 + (1+x^2)\sec^2(x-y)}Therefore: dxdy=1+(1+x2)sec2(x−y)(1+x2)sec2(x−y)+2xytan(x−y)