## Exercises - Calculating Derivative: Inverse, Implicit and Parametric
Functions

We start with problems on derivative of inverse functions, then do some on
implicit functions and differentiation
including tangent lines. Next we ask about derivatives and tangent lines for
parametric curves and as a bonus we look
at some
angles of intersection.

If you want to refer to sections of Methods Survey on implicit and parametric
functions while working the exercises, you can click
here and it will appear in a
separate full-size window. Similarly,
here we offer Theory.

For each of the following functions *f* and a given point *b*, guess
some *a* such that
*f* (*a*) = *b* and then prove that
*f* is 1-1 on some neighborhood of *a* and therefore it has an
inverse function *f*_{−1} there. Then find
( *f*_{−1})′(*b*), the derivative of
this inverse at *b*.

For each of the following curves, show that it can be given as a graph of
some function
*y* = *y*(*x*) on a neighborhood of
the given point
(*a*,*b*). Then use implicit differentiation to find
*y*′(*a*) and
*y*′′(*a*) and find the tangent line to this curve at
(*a*,*b*).

For each of the following parametric curves, find the tangent line and the
normal line at the indicated point.

The following parametric curves can be locally expressed as graphs of
functions
*y* = *y*(*x*).
Find the derivatives
*y*′(*x*) and
*y*′′(*x*).

Bonus: Find the angles at which the following pairs of curves intersect.

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