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In mathematics, the concept of a derivative is fundamental for understanding rates of change. The derivative can be seen as a tool that describes how a function behaves as its input values change. This is often visualized as the slope of the tangent line to the function at a given point.

The formal definition of a derivative is given by:

• \[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]

Here are the steps to find a derivative using this definition:

• Find \(f(x + h)\) by substituting \(x + h\) into the function.
• Calculate the difference quotient \(\frac{f(x+h) - f(x)}{h}\).
• Simplify the expression to make taking the limit as \(h\) approaches 0 easier.
• Evaluate the limit to find the derivative \(f'(x)\).

Using the formal definition of the derivative helps us understand exactly how we mathematically compute the slope of the tangent line to a function, which is key for deeper insights into its behavior.

The tangent line is a straight line that touches a curve at exactly one point, representing the best linear approximation to the curve at that point. This line gives us the slope of the function at that specific location. For the function \(y = \frac{1}{x}\) at \(x = 2\), it played a crucial role in understanding how the function changes around that point.

To find the equation of the tangent line at a point:

• First, compute the derivative of the function at that point. For instance, if we have the derivative \(f'(2) = -\frac{1}{4}\), this value is the slope of the tangent line.
• Use the point-slope form: \[y - y_1 = m(x - x_1)\] where \(m\) is the slope (\(-\frac{1}{4}\)) and \((x_1, y_1)\) is the point \((2, \frac{1}{2})\).

Plugging the values, we get the equation of the tangent line as:

• \[y - \frac{1}{2} = -\frac{1}{4}(x - 2)\]

Simplifying further yields the exact linear equation representing the tangent line at that specific point.

The normal line to a curve at a given point is perpendicular to the tangent line at that same point. Understanding the normal line is important because it helps in analyzing how the function's graph is bending at a certain location.

If the slope of the tangent line is known, say \(-\frac{1}{4}\), the normal line's slope is the negative reciprocal of this slope. This is calculated as follows:

• Slope of normal line = \(-\frac{1}{\left(-\frac{1}{4}\right)} = 4\)

Using the point-slope form once again, the equation for the normal line at point \((2, \frac{1}{2})\) becomes:

• \[y - \frac{1}{2} = 4(x - 2)\]

Simplifying, we get:

• \[y = 4x - 7.5\]

This gives an equation for the normal line, showcasing its perpendicular relationship with the tangent line and highlighting a key feature of the curve at the point \((2, \frac{1}{2})\).