Find the equations of the tangent lines to the following...

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Key Concepts

Handling Vertical Tangents

In some situations, especially with implicitly defined curves, the derivative may become undefined at certain points, indicating a vertical tangent line. In such cases, the tangent line cannot be expressed in the form y = mx + b, and instead is written as x = constant. This concept is crucial when evaluating the behavior of the curve at points where the standard derivative formula leads to a division by zero.

Implicit Differentiation

When dealing with equations where y is not isolated, implicit differentiation is used to differentiate both sides of the equation with respect to x while treating y as a function of x. This process often involves applying the chain rule to terms that include y, and then solving for dy/dx to find the slope of the tangent line.

Tangent Line Equation

The tangent line to a curve at a given point is the straight line that best approximates the curve near that point. Its slope is given by the derivative evaluated at that point, and its equation can be written using the point-slope form: y - y0 = m(x - x0), where (x0, y0) is the point of tangency and m is the slope.

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