4. Applications of Derivatives

Vertical tangent lines

a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).

Given the equation $x{e}^y=x-y$, find $\frac{dy}{dx}$ using implicit differentiation.

Parallel tangent lines Find the two points where the curve x² + xy + y² = 7 crosses the x-axis, and show that the tangent lines to the curve at these points are parallel. What is the common slope of these tangent lines?

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

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x + √xy = 6, (4, 1)