SUNY Geneseo Department of Mathematics
Math 221 02
Fall 2020
Prof. Doug Baldwin
This discussion is a place to practice, and share ideas on, implicit differentiation. It’s based on section 3.8 in our textbook. For now I’ve posed some questions that practice the basic idea of implicit differentiation; in a couple of days I expect to extend this with additional questions involving applications or more advanced aspects of it.
Find dy/dx, given that x and y are related by the equation
\[3x^2 - 5y^2 = 1\]Find dy/dt, given that y and t are related by the equation
\[t^2 - 2ty + y = 0\]Finally, find dz/dx given that z and x are related by the equation
\[\cos(z^2+x^2) = 1\]Adding some more complicated uses of implicit differentiation, see if you can use it to find dy/dx given that...
\[e^{xy} = 1\]or
\[\frac{x}{x-y} = x + y\]For either of the two equations above, can you also find dx/dy? Does dx/dy even make sense at all as a thing to find?
And finally, setting the stage for talking about implicit differentiation and derivatives of inverses of functions, could you find dy/dx if
\[y = \arcsin x = \sin^{-1}x\]