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A derivative represents the rate of change of a function with respect to a variable. In simple terms, it's like saying how fast the output of the function is changing at any given point. Calculating the derivative is crucial when you want to find the slope of the tangent line to the graph of the function.
For the function given in the exercise, which is \(f(x) = 5x - x^2\), we need to find its derivative with respect to \(x\). The derivative, represented as \(f'(x)\), is calculated by differentiating each term. The derivative of \(5x\) is 5, and the derivative of \(-x^2\) is \(-2x\). Combining these results, we get \(f'(x) = 5 - 2x\).
This derivative function helps us determine the slope of the tangent at any given point on the graph of the function.

The slope of the tangent line is a measure of how steep the line is at a specific point on a curve. To find this slope, you use the derivative of the function. Remember, the derivative gives us the rate of change of the function, which corresponds to the slope of the tangent.

• We already have the derivative \(f'(x) = 5 - 2x\).
• To find the slope of the tangent line at the point \((c, f(c))\), substitute \(c = 4\) into the derivative.

For our function, this means evaluating \(f'(4) = 5 - 2(4) = -3\).
So, the slope of the tangent line at the point \((4, 4)\) is \(-3\). This negative value indicates that the tangent line decreases as it moves from left to right.

The point-slope form is a linear equation format that is especially useful for finding the equation of a line when a point and the slope of the line are known. The general formula is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point on the line and \(m\) is the slope.
To apply this to the exercise, identify your point as \((4, 4)\) and your slope as \(-3\). Plug these values into the formula to get the equation of the tangent line:

• Substitute: \(y - 4 = -3(x - 4)\)
• Simplify if necessary, but in this case, the form is acceptable as it defines the tangent line explicitly.

The point-slope form efficiently encapsulates both the location and the direction (slope) of the tangent line, making it a powerful tool in calculus for describing lines.