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A tangent line is a straight line that just touches a curve at a given point and has the same slope as the curve does at that point. This means it's the best linear approximation of the curve at that specific location. In order to find the equation of the tangent line, we need two key pieces of information:

• The slope of the tangent line at the point of tangency
• The coordinates of the point where the curve and line meet

For the given problem, we've calculated the slope of the tangent line by using implicit differentiation, which gave us a slope of \( \frac{7}{2} \) at the point \((1, 2)\). Once we know the slope, we can use the point-slope form of a line, \( y - y_1 = m(x - x_1) \), where \(m\) is the slope and \((x_1, y_1)\) is the point of tangency. Substituting the slope and coordinates into this formula enables us to derive the equation of the tangent line: \[y = \frac{7}{2}x - \frac{3}{2}\]This equation describes the tangent line that touches the curve at the point \((1, 2)\). Here, the curve is our hyperbola, and this tangent line reflects the curve's behavior right at that instant.

The product rule is a fundamental technique for differentiation when dealing with products of two functions. It's essential for problems like the one you're working on, where terms involve the product of variables. In mathematical terms, if you have two differentiable functions, \( u(x) \) and \( v(x) \), the product rule states:\[(uv)' = u'v + uv'\]This is used when differentiating the term \( 2xy \) in the problem. This term is a product of the functions \( u(x) = 2x \) and \( v(y) = y \). Applying the product rule gives us:

• Differentiate \( 2x \) with respect to \( x \) to get \( 2 \)
• Keep \( y \) as it is to get \( 2 \cdot y \)
• Keep \( 2x \) as it is and differentiate \( y \) with respect to \( x \), which involves using implicit differentiation, to get \( 2x \cdot \frac{dy}{dx} \)

Combine these to get:\[2xy' + 2y\]This breakdown shows how the product rule allows us to handle such terms, where each part contributes to the full derivative. It's a nifty tool that’s a must-know for calculus, especially when the expressions involve multiple variables like in your hyperbola problem.

A hyperbola is a type of conic section, shaped like an open curve with two branches. Every hyperbola has a specific equation form, typically expressed as \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). It can seem complex at first sight, but this conic section needs analysis to discover its unique properties. The equation given in this problem is indeed a hyperbola: \[x^2 + 2xy - y^2 + x = 2\]To find the tangent line at a particular point on this hyperbola, we follow the implicit differentiation method, as direct differentiation isn't feasible due to \( x \) and \( y \) being intertwined. Hyperbolas have the distinct property of approaching two fixed lines, known as asymptotes, yet never intersecting them.
These asymptotes can be seen as tangent lines that the hyperbola gets infinitely close to but never quite touches. This nature of hyperbolas means that the tangent line we derived in this problem also follows similar calculations.In essence, understanding the nature of a hyperbola helps appreciate the linked concepts of curves, tangents, and implicit differentiation in calculus. When you grasp these connections, tackling such problems becomes more manageable and less daunting.