Assume that $x$ and $y$ are both differentiable functions of $t$ and find the required variables below.
Equation: $x^{2}+y^{2}=9$
Use implicit differentiation $\frac{d y}{d t}=$ (value)
a. Find $\frac{d y}{d t}$ when $x=2$ and $y=\sqrt{5}$ given $\frac{d x}{d t}=-1$ $\frac{d y}{d t}=\sqrt{\frac{2}{\sqrt{5}}} \quad$ (value)
Answer
Simplifying this, we get \(\frac{dy}{dt} = \boxed{\sqrt{\frac{2}{\sqrt{5}}}}\).
Step 1 ：Given the equation \(x^{2}+y^{2}=9\), we can differentiate both sides with respect to \(t\) to get \(2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0\).
Step 2 ：We can simplify this to \(x\frac{dx}{dt} + y\frac{dy}{dt} = 0\).
Step 3 ：Rearranging for \(\frac{dy}{dt}\), we get \(\frac{dy}{dt} = -\frac{x}{y}\frac{dx}{dt}\).
Step 4 ：Substituting the given values \(x=2\), \(y=\sqrt{5}\), and \(\frac{dx}{dt}=-1\) into the equation, we get \(\frac{dy}{dt} = -\frac{2}{\sqrt{5}}(-1)\).
Step 5 ：Simplifying this, we get \(\frac{dy}{dt} = \boxed{\sqrt{\frac{2}{\sqrt{5}}}}\).
