if $\\sqrt{x}+\\sqrt{y}=11$ and $y(25)=36$, find $y(25)$ by implicit differentiation.
if $\\sqrt{x}+\\sqrt{y}=11$ and $y(25)=36$, find $y(25)$ by implicit differentiation.
Answer
Explanation:
Step1: Differentiate both sides of the equation
Differentiate (\sqrt{x}+\sqrt{y}=11) with respect to (x). Using the power rule ((x^n)^\prime = nx^{n - 1}), we have (\frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{y}}y^\prime=0).
Step2: Solve for (y^\prime)
Multiply through by (2) to get (\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}y^\prime = 0). Then isolate (y^\prime): (\frac{1}{\sqrt{y}}y^\prime=-\frac{1}{\sqrt{x}}), so (y^\prime=-\frac{\sqrt{y}}{\sqrt{x}}).
Step3: Substitute (x = 25) and (y=36)
When (x = 25) and (y = 36), (y^\prime=-\frac{\sqrt{36}}{\sqrt{25}}).
Answer:
(-\frac{6}{5})