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To solve problems involving differentials, a critical first step is to find the derivative of the given function. Derivatives measure how a function changes as its input changes, and are a fundamental part of differential calculus. To evaluate the derivative of a polynomial function like \( y = x^2 - 5x - 6 \), you need to differentiate each term of the expression. Differentiation rules tell us:

• The derivative of \( x^n \) is \( nx^{n-1} \).
• Constant terms, such as \(-6\), have a derivative of 0 because they don't change with \( x \).

So, applying these rules, the derivative \( \frac{dy}{dx} \) of \( y = x^2 - 5x - 6 \) becomes \( 2x - 5 \). This computed derivative now describes the instantaneous rate of change of the function at any point \( x \).

Once the derivative is calculated, it's time to construct a differential equation. Differential equations link derivatives and differentials to show how a small change in one variable affects the change in another variable. For our function, the differential is expressed using the derivative: \[ dy = \frac{dy}{dx} \times dx \]This formula states that the small change in \( y \), denoted as \( dy \), is equal to the derivative \( \frac{dy}{dx} \) multiplied by a small change in \( x \), represented by \( dx \). Substituting the derivative \( 2x - 5 \) into the formula, we get:\[ dy = (2x - 5) \cdot dx \]This expression lets us approximate how much \( y \) changes when \( x \) changes by a small amount \( dx \). Differential equations are crucial tools for modeling dynamic systems and understanding how quantities interrelate.

Tangent line approximation leverages the derivative to approximate changes in the function around a point. Imagine a curve on a graph; the tangent line is a straight line that just "touches" the curve at a single point. The slope of this line is given by the derivative at that point. In our given problem, we're finding \( dy \) at \( x = 2 \) with \( dx = 0.1 \). This involves evaluating \[ dy = (2 \cdot 2 - 5) \cdot 0.1 = (-1) \cdot 0.1 = -0.1 \]This means that around \( x = 2 \), if \( x \) changes by 0.1, we can expect \( y \) to decrease by 0.1. The approximation is valid for small values of \( dx \) and gives insight into the function's behavior nearby without extensive calculations. Tangent lines provide a linear approach to understanding nonlinear functions in the local vicinity of a given point.