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When we talk about derivatives in calculus, we are discussing a fundamental concept that measures how a function changes as its input changes. This is akin to examining how quickly a car is moving at a specific moment if you think of the function as a path the car is driving along.

The formal definition of the derivative of a function at a point, represented as \( f'(x) \), is the limit of the average change (or difference quotient) as the interval considered approaches zero. For our example where the function is \( y = x^3 - 9x \), the derivative \( y' = 3x^2 - 9 \) represents the slope of the curve at any point \( x \). In simpler terms, if you were walking along the graph of the function, the derivative tells you exactly how steep the hill is at every step.

Imagine you're taking a leisurely hike up a mountain, the path ahead of you twisting and turning. The slope of the curve in calculus resembles the incline or decline you experience on such a hike. It describes the direction and steepness of the curve at any given point.

Mathematically speaking, the slope at a specific point on a curve is the value of the derivative at that point. Therefore, when we calculated the derivative \( y' = 3x^2 - 9 \) for the function \( y = x^3 - 9x \) and found the slope \( m \) at \( x = 1 \), we discovered how steep the curve is at that precise location. On a graph, it shows how fast the \( y \) value is changing compared to the \( x \) value. The slope is a crucial piece of information because, in the context of finding tangent lines, it gives us the exact angle at which the tangent line should leave the curve.

Sometimes, you need to pin a line to a specific point, like pinning a note to a bulletin board. The point-slope form does just that for lines on a graph. It is an equation of a line that includes both the line's slope and its intersection at a particular point. Expressed as \( y - y1 = m(x - x1) \) where \( m \) stands for the slope and \( (x1, y1) \) is a given point through which the line passes.

For our initial problem, once we have the slope \( m \) from the function's derivative at \( x = 1 \) and our point (1, -9), we plug these into the point-slope equation. Algebra then takes the wheel, allowing us to isolate \( y \) to find the line's equation in slope-intercept form (\( y = mx + b \) ). This makes plotting the line straightforward as it reveals both the incline and the starting point of the line on the \( y \) axis, serving as a 'map' to draw the line directly on the graph.