Implicit differentiation

asimon2005

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Find dy/dx by implicit differentiation.

14. Y sin(x^2) = x sin(y^2)

18. Tan(x-y) = y/1+x^2
 
Hello, asimon2005!

Find dydx by implicit differentiation..\displaystyle \text{Find }\frac{dy}{dx}\text{ by implicit differentiation..}

. . 14.    ysin(x2)=xsin(y2)\displaystyle 14.\;\;y\cdot\sin(x^2) \:=\: x\cdot\sin(y^2)

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)

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)

Rearrange terms:   sin(x2)dydx2xycos(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)

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)


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)}





We have:   (1+x2)tan(xy)  =  y\displaystyle \text{We have: }\;(1+x^2)\cdot\tan(x-y) \;=\;y

Product Rule:   (1+x2) ⁣ ⁣sec2(xy) ⁣ ⁣(1dydx)+(2x) ⁣ ⁣tan(xy)  =  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}

We have:   (1+x2) ⁣ ⁣sec2(xy)(1+x2) ⁣ ⁣sec2(xy)dydx+2x ⁣ ⁣tan(xy)  =  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}

Rearrange terms:   dydx  +  (1+x2)sec2(xy)dydx  =  (1+x2)sec2(xy)  +  2xtan(xy)\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)

Factor:   [1+(1+x2)sec2(xy)]dydx  =  (1+x2)sec2(xy)+2xytan(xy)\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)


Therefore:   dydx  =  (1+x2)sec2(xy)+2xytan(xy)1+(1+x2)sec2(xy)\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)}

 
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