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Implicit differentiation is a powerful technique used when dealing with equations where the variables are mixed together, rather than explicitly solved for one variable. For instance, the equation given in the problem, \( y^2 - 4x^2 = 5 \), is not solved for \( y \) in terms of \( x \). This is common with hyperbolas and other complex curves. Implicit differentiation allows us to find the derivative without explicitly solving for one of the variables first. Here’s how it works:

• Differentiate both sides of the equation with respect to \( x \) as if \( y \) is a function of \( x \).
• Apply the chain rule for terms involving \( y \), treating \( y \) as \( y(x) \).

This is why when differentiating \( y^2 \), we use the chain rule, resulting in \( 2y \frac{dy}{dx} \). This keeps \( dy/dx \) in the expression, helping us solve for it later. The technique essentially recognizes the dependence of \( y \) on \( x \) without needing an explicit function, which saves a lot of algebraic manipulation.Implicit differentiation is used frequently in calculus, especially when dealing with more complex shapes and equations. It simplifies finding derivatives where explicit differentiation would be cumbersome or impossible.

A hyperbola is one of the basic types of conic sections, defined by its distinct shape and properties. Hyperbolas appear like two mirrored, open curves. The standard form of a hyperbola's equation is \( x^2/a^2 - y^2/b^2 = 1 \) or \( y^2/b^2 - x^2/a^2 = 1 \). Our given equation, \( y^2 - 4x^2 = 5 \), after some manipulation, fits this structure.Hyperbolas have unique characteristics:

• They are defined by two branches, each located on opposite sides of two lines called asymptotes.
• Each branch approaches these asymptotes but never intersects them.
• The center, traversed along the major axis, defines a symmetry through which hyperbola properties are symmetric.

In the context of calculus, understanding the geometric nature of hyperbolas helps us determine where tangent lines meet their curves. This relationship is crucial when calculating slopes or determining intersection angles within applied problems.

Derivative calculation is essential for finding the slope of the tangent line at a specific point on a curve. In this exercise, we begin by implicitly differentiating the equation \( y^2 - 4x^2 = 5 \), focusing on how each component evolves with respect to \( x \). During differentiation:

• For \( y^2 \), applying the chain rule yields \( 2y \frac{dy}{dx} \), indicating how \( y \) changes as \( x \) changes.
• The term \( -4x^2 \) directly differentiates to \( -8x \) since it's explicitly in terms of \( x \).
• The derivative of the constant \( 5 \) is 0, simplifying the differentiation process.

After rearranging and solving the differential equation \( 2y \frac{dy}{dx} = 8x \), we isolate \( \frac{dy}{dx} \) to find the expression \( \frac{dy}{dx} = \frac{4x}{y} \). This gives us a general formula for the slope of the tangent line for any point \( (x,y) \) on the hyperbola. Finally, substituting the point \( P(-1, 3) \) into this formula, we find that \( \frac{dy}{dx} \) equals \( \frac{-4}{3} \), providing the slope at that exact point. Calculating derivatives not only helps find slopes but is a fundamental step in understanding the rates of change within any function.