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To find the tangent line to a curve at a particular point, we need to understand the concept of a derivative. The derivative of a function gives us the slope of the tangent line at any point on the curve. It tells us how steep the curve is at that point. For the function given in the exercise, \( y = x^3 - x \), the derivative is \( y' = 3x^2 - 1 \). This formula allows us to calculate the slope of the curve at any \( x \) value.

The process of finding the derivative involves using rules like the power rule, where the derivative of \( x^n \) is \( nx^{n-1} \). This is fundamental for calculus and helps us understand not just where a curve is heading, but how fast it changes at any given point.

Once we have the derivative, we use it to find the slope of the curve at a specific point. In the exercise, we calculate the slope where \( x = -1 \). Substituting \( -1 \) into the derivative \( y' = 3x^2 - 1 \) gives \( y'(-1) = 3(-1)^2 - 1 = 2 \).

This result, a slope of 2, means that at \( x = -1 \), the curve rises by 2 units vertically for every 1 unit it moves horizontally. Understanding this slope is crucial in determining how the tangent line behaves at the point, giving us a clear picture of the curve's behavior in that small region.

To write the equation of the tangent line, we use the point-slope form, which is \( y - y_1 = m(x - x_1) \). Here, \( m \) is the slope of the line and \( (x_1, y_1) \) is a point on the line. For this exercise, with slope \( m = 2 \) at point \( (-1, 0) \), it becomes \( y - 0 = 2(x + 1) \).

Simplifying, we get \( y = 2x + 2 \). This line equation shows how the tangent will touch the curve exactly at \( (-1,0) \), maintaining the slope of 2. Writing equations of lines through different points forms the backbone of geometry and calculus, allowing us to approach problems involving tangent lines, intersections, and more.

Finding where a tangent line intersects a curve or another line involves solving equations simultaneously. For our exercise, solving \( x^3 - x = 2x + 2 \) results in \( x^3 - 3x - 2 = 0 \).

This equation needs to be solved to discover the x-coordinates of intersection points. Using techniques such as factoring or solver tools reveals solutions like \( x = -1 \) and \( x = 0.5 \).

Each solution x-value corresponds to a point where the tangent line intersects the curve. Checking the equation for these x-values confirms the point coordinates, such as \( (0.5, 0) \). These points of intersection bring critical insights when examining functions and their tangents, making them essential in various applications.