exercises 3.8 implicit differentiation\nscore: 8/20 answered: 4/10\nprogress saved done\nquestion 5\n0/2 pts…

exercises 3.8 implicit differentiation\nscore: 8/20 answered: 4/10\nprogress saved done\nquestion 5\n0/2 pts 100 99 details\ntextbook videos +\ngiven \\( \\sqrt { x } + \\sqrt { y } = 11 \\)\nfind \\( \\frac { d y } { d x } \\) by implicit differentiation. write it down!\nthe point \\( ( 25,36 ) \\) is on the graph. find \\( y ^ { \\prime } ( 25 ) \\) or the slope\nof the tangent line at \\( x = 25 \\).\nquestion help: video message instructor\nsubmit question jump to answer

exercises 3.8 implicit differentiation\nscore: 8/20 answered: 4/10\nprogress saved done\nquestion 5\n0/2 pts 100 99 details\ntextbook videos +\ngiven \\( \\sqrt { x } + \\sqrt { y } = 11 \\)\nfind \\( \\frac { d y } { d x } \\) by implicit differentiation. write it down!\nthe point \\( ( 25,36 ) \\) is on the graph. find \\( y ^ { \\prime } ( 25 ) \\) or the slope\nof the tangent line at \\( x = 25 \\).\nquestion help: video message instructor\nsubmit question jump to answer

Answer

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (\sqrt{x}+\sqrt{y}=11). Using the power rule ((x^n)^\prime = nx^{n - 1}) and the chain - rule ((y^n)^\prime=n y^{n - 1}\frac{dy}{dx}). (\frac{d}{dx}(\sqrt{x})+\frac{d}{dx}(\sqrt{y})=\frac{d}{dx}(11)) (\frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{y}}\frac{dy}{dx}=0)

Step2: Solve for (\frac{dy}{dx})

Subtract (\frac{1}{2\sqrt{x}}) from both sides: (\frac{1}{2\sqrt{y}}\frac{dy}{dx}=-\frac{1}{2\sqrt{x}}) Multiply both sides by (2\sqrt{y}) to get (\frac{dy}{dx}=-\frac{\sqrt{y}}{\sqrt{x}})

Step3: Evaluate (\frac{dy}{dx}) at the point ((25,36))

Substitute (x = 25) and (y = 36) into (\frac{dy}{dx}=-\frac{\sqrt{y}}{\sqrt{x}}) (\frac{dy}{dx}\big|_{x = 25,y = 36}=-\frac{\sqrt{36}}{\sqrt{25}})

Answer:

(-\frac{6}{5})