Question

exercises 3.8 implicit differentiation
score: 8/20 answered: 4/10
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question 5
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given \\( \sqrt { x } + \sqrt { y } = 11 \\)
find \\( \frac { d y } { d x } \\) by implicit differentiation. write it down!
the point \\( ( 25,36 ) \\) is on the graph. find \\( y ^ { \prime } ( 25 ) \\) or the slope
of the tangent line at \\( x = 25 \\).
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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:

\(-\frac{6}{5}\)