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3.25 Examples of implicit differentiation

The formula $x^{2}+4y^{2}=1$ defines an ellipse. Find ${{dy}\over{dx}}.$ (See diagram.)

Solution.We have $f(x,y)=x^{2}+4y^{2}-1$ , so that $f(x,y)=0$ gives the ellipse. Now differentiate with respect to $x$ , to get:

\frac{d}{dx}f(x,y)=\,{2x+8y\frac{dy}{dx}=0.}

Thus $8y\frac{dy}{dx}=\,{-2x,}$ so $\frac{dy}{dx}=\,{-\frac{x}{4y}.}$

Note that we can parametrize the curve via $x=\cos t$ , $y=\frac{1}{2}\sin t$ and then

\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\,{\frac{\frac{1}{2}\cos t}{-\sin t}}\,{=-% \frac{\cos t}{4\cdot\frac{1}{2}\sin t}=-\frac{x}{4y}}

as claimed.