Problems

1. 1. Differentiate implicitly to find the slope of the tangent line to the curve `x=y^2` at `x=1` and `y=1`.
   2. Calculate `dytext[/]dx` for the equation `x=y^2`, and use the derivative to find the slope of the tangent line to this curve at `text[(]1,1text[)]`.
   3. Find the slope of the tangent line to the curve `x=y^2` at `text[(]1,1text[)]` by first solving for `y` as an explicit function of `x`.
2. Use the inverse function derivative formula to give another derivation of the formula

   `d/(dx) sqrt(x)=1/(2sqrt(x))`.

3. 1. Use implicit differentiation to obtain a formula for the derivative of `x^(1//3)`.
   2. Use the inverse function derivative formula to confirm your result in part (a).
4. We began our discussion of implicit differentiation by finding a slope formula for points on the ellipse `x^2`/`4+y^2`/`9=1`:

   `(dy)/(dx)=-(9x)/(4y)`.

   We applied the formula only to points on the upper half of the ellipse.

   1. Find the slope at the point on the lower half of the ellipse at which `x=1`.
   2. Find an explicit formula for slopes at all points on the lower half of the ellipse.
5. In Checkpoint 2 you found slopes on the upper half of the hyperbola `y^2-x^2=1`.
   1. Find the slope at the point on the lower branch of the hyperbola at which `x=1`.
   2. Find an explicit formula for slopes at all points on the lower branch of the hyperbola.