Derivada implícita : raiz (x+y) + raiz(x-y)=6
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Derivando os dois lados da expressão em relação a x:
![\sqrt{x+y}+\sqrt{x-y}=6\\\\
\dfrac{d}{dx}(\sqrt{x+y}+\sqrt{x-y})=\dfrac{d}{dx}(6)\\\\
\dfrac{d}{dx}(\sqrt{x+y})+\dfrac{d}{dx}(\sqrt{x-y})=0\\\\
\dfrac{d}{dx}[(x+y)^{\frac 1 2}]+\dfrac{d}{dx}[(x-y)^{\frac 1 2}]=0\\\\
~[\frac 1 2 \cdot (x+y)^{-\frac 1 2}\cdot(1+y')]+[\frac 1 2 \cdot (x-y)^{-\frac 1 2}\cdot(1-y')]=0\\\\
~[(x+y)^{-\frac 1 2}\cdot(1+y')]+[(x-y)^{-\frac 1 2}\cdot(1-y')]=0\\\\
(x+y)^{-\frac 1 2}+y'(x+y)^{-\frac 1 2}+(x-y)^{-\frac 1 2}-y'(x-y)^{-\frac 1 2}=0](https://tex.z-dn.net/?f=%5Csqrt%7Bx%2By%7D%2B%5Csqrt%7Bx-y%7D%3D6%5C%5C%5C%5C%0A%5Cdfrac%7Bd%7D%7Bdx%7D%28%5Csqrt%7Bx%2By%7D%2B%5Csqrt%7Bx-y%7D%29%3D%5Cdfrac%7Bd%7D%7Bdx%7D%286%29%5C%5C%5C%5C%0A%5Cdfrac%7Bd%7D%7Bdx%7D%28%5Csqrt%7Bx%2By%7D%29%2B%5Cdfrac%7Bd%7D%7Bdx%7D%28%5Csqrt%7Bx-y%7D%29%3D0%5C%5C%5C%5C%0A%5Cdfrac%7Bd%7D%7Bdx%7D%5B%28x%2By%29%5E%7B%5Cfrac+1+2%7D%5D%2B%5Cdfrac%7Bd%7D%7Bdx%7D%5B%28x-y%29%5E%7B%5Cfrac+1+2%7D%5D%3D0%5C%5C%5C%5C%0A%7E%5B%5Cfrac+1+2+%5Ccdot+%28x%2By%29%5E%7B-%5Cfrac+1+2%7D%5Ccdot%281%2By%27%29%5D%2B%5B%5Cfrac+1+2+%5Ccdot+%28x-y%29%5E%7B-%5Cfrac+1+2%7D%5Ccdot%281-y%27%29%5D%3D0%5C%5C%5C%5C%0A%7E%5B%28x%2By%29%5E%7B-%5Cfrac+1+2%7D%5Ccdot%281%2By%27%29%5D%2B%5B%28x-y%29%5E%7B-%5Cfrac+1+2%7D%5Ccdot%281-y%27%29%5D%3D0%5C%5C%5C%5C%0A%28x%2By%29%5E%7B-%5Cfrac+1+2%7D%2By%27%28x%2By%29%5E%7B-%5Cfrac+1+2%7D%2B%28x-y%29%5E%7B-%5Cfrac+1+2%7D-y%27%28x-y%29%5E%7B-%5Cfrac+1+2%7D%3D0 "\sqrt{x+y}+\sqrt{x-y}=6\\\\
\dfrac{d}{dx}(\sqrt{x+y}+\sqrt{x-y})=\dfrac{d}{dx}(6)\\\\
\dfrac{d}{dx}(\sqrt{x+y})+\dfrac{d}{dx}(\sqrt{x-y})=0\\\\
\dfrac{d}{dx}[(x+y)^{\frac 1 2}]+\dfrac{d}{dx}[(x-y)^{\frac 1 2}]=0\\\\
~[\frac 1 2 \cdot (x+y)^{-\frac 1 2}\cdot(1+y')]+[\frac 1 2 \cdot (x-y)^{-\frac 1 2}\cdot(1-y')]=0\\\\
~[(x+y)^{-\frac 1 2}\cdot(1+y')]+[(x-y)^{-\frac 1 2}\cdot(1-y')]=0\\\\
(x+y)^{-\frac 1 2}+y'(x+y)^{-\frac 1 2}+(x-y)^{-\frac 1 2}-y'(x-y)^{-\frac 1 2}=0")
![y'[(x-y)^{-\frac 1 2}-(x+y)^{-\frac 1 2}]=(x+y)^{-\frac 1 2}+(x-y)^{-\frac 1 2}\\\\
y'=\dfrac{(x+y)^{-\frac 1 2}+(x-y)^{-\frac 1 2}}{(x-y)^{-\frac 1 2}-(x+y)^{-\frac 1 2}}\\\\
y'=\dfrac{\dfrac{1}{\sqrt{x+y}}+\dfrac{1}{\sqrt{x-y}}}{\dfrac{1}{\sqrt{x-y}}-\dfrac{1}{\sqrt{x+y}}}\\\\\\
y'=\dfrac{~~~\dfrac{\sqrt{x+y}+\sqrt{x-y}}{\sqrt{x-y}\cdot\sqrt{x+y}}~~~}{\dfrac{\sqrt{x+y}-\sqrt{x-y}}{\sqrt{x-y}\cdot\sqrt{x+y}}}\\\\\\
y'=\dfrac{\sqrt{x+y}+\sqrt{x-y}}{\sqrt{x+y}-\sqrt{x-y}}](https://tex.z-dn.net/?f=y%27%5B%28x-y%29%5E%7B-%5Cfrac+1+2%7D-%28x%2By%29%5E%7B-%5Cfrac+1+2%7D%5D%3D%28x%2By%29%5E%7B-%5Cfrac+1+2%7D%2B%28x-y%29%5E%7B-%5Cfrac+1+2%7D%5C%5C%5C%5C%0Ay%27%3D%5Cdfrac%7B%28x%2By%29%5E%7B-%5Cfrac+1+2%7D%2B%28x-y%29%5E%7B-%5Cfrac+1+2%7D%7D%7B%28x-y%29%5E%7B-%5Cfrac+1+2%7D-%28x%2By%29%5E%7B-%5Cfrac+1+2%7D%7D%5C%5C%5C%5C%0Ay%27%3D%5Cdfrac%7B%5Cdfrac%7B1%7D%7B%5Csqrt%7Bx%2By%7D%7D%2B%5Cdfrac%7B1%7D%7B%5Csqrt%7Bx-y%7D%7D%7D%7B%5Cdfrac%7B1%7D%7B%5Csqrt%7Bx-y%7D%7D-%5Cdfrac%7B1%7D%7B%5Csqrt%7Bx%2By%7D%7D%7D%5C%5C%5C%5C%5C%5C%0Ay%27%3D%5Cdfrac%7B%7E%7E%7E%5Cdfrac%7B%5Csqrt%7Bx%2By%7D%2B%5Csqrt%7Bx-y%7D%7D%7B%5Csqrt%7Bx-y%7D%5Ccdot%5Csqrt%7Bx%2By%7D%7D%7E%7E%7E%7D%7B%5Cdfrac%7B%5Csqrt%7Bx%2By%7D-%5Csqrt%7Bx-y%7D%7D%7B%5Csqrt%7Bx-y%7D%5Ccdot%5Csqrt%7Bx%2By%7D%7D%7D%5C%5C%5C%5C%5C%5C%0Ay%27%3D%5Cdfrac%7B%5Csqrt%7Bx%2By%7D%2B%5Csqrt%7Bx-y%7D%7D%7B%5Csqrt%7Bx%2By%7D-%5Csqrt%7Bx-y%7D%7D "y'[(x-y)^{-\frac 1 2}-(x+y)^{-\frac 1 2}]=(x+y)^{-\frac 1 2}+(x-y)^{-\frac 1 2}\\\\
y'=\dfrac{(x+y)^{-\frac 1 2}+(x-y)^{-\frac 1 2}}{(x-y)^{-\frac 1 2}-(x+y)^{-\frac 1 2}}\\\\
y'=\dfrac{\dfrac{1}{\sqrt{x+y}}+\dfrac{1}{\sqrt{x-y}}}{\dfrac{1}{\sqrt{x-y}}-\dfrac{1}{\sqrt{x+y}}}\\\\\\
y'=\dfrac{~~~\dfrac{\sqrt{x+y}+\sqrt{x-y}}{\sqrt{x-y}\cdot\sqrt{x+y}}~~~}{\dfrac{\sqrt{x+y}-\sqrt{x-y}}{\sqrt{x-y}\cdot\sqrt{x+y}}}\\\\\\
y'=\dfrac{\sqrt{x+y}+\sqrt{x-y}}{\sqrt{x+y}-\sqrt{x-y}}")
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