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When studying calculus, the concept of a derivative is fundamental. A derivative represents the rate at which a function changes at any point. It's like the speedometer in your car, showing how fast something changes. In formal terms, if you have a function \( f(x) \), its derivative \( f'(x) \) tells you the slope of the tangent line at any \( x \) value.

For example, the function given here is \( f(x) = \frac{1}{x} - x^{2/3} \). To find its derivative \( f'(x) \), you must understand how each part of \( f(x) \) behaves when differentiated. For \( \frac{1}{x} \), the derivative is \( -\frac{1}{x^2} \). The term \( x^{2/3} \) involves using the power rule, which we will touch on later. Together, they form the overall derivative:

• \( -\frac{1}{x^2} \) from \( \frac{1}{x} \)
• \( -\frac{2}{3}x^{-1/3} \) from \( -x^{2/3} \)

So, \( f'(x) = -\frac{1}{x^2} - \frac{2}{3}x^{-1/3} \). This captures how \( f(x) \) changes at each point \( x \). Derivatives are crucial in many fields beyond mathematics, like physics, engineering, and economics.

The concept of a tangent line is closely linked to derivatives. Imagine a straight line that just "touches" a curve at a single, specific point, without crossing it. This is the tangent line, and it shares the same slope as the curve at the point of contact.

To find the equation of a tangent line to a function at a specific \( x \) value, we use the derivative. It's crucial because the derivative gives us the slope of the function at that point. In our exercise, we calculated the slope, which is \( f'(-1) = -\frac{5}{3} \). Using this in the point-slope form equation of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is our slope and \( (x_1, y_1) \) is the point of tangency, \( (-1, -2) \), we have:

• \( y + 2 = -\frac{5}{3}(x + 1) \)
• Simplified: \( y = -\frac{5}{3}x - \frac{11}{3} \)

This is the equation of the tangent line. You can visualize it as a straight line barely skimming the curve at \( x = -1 \). This helps us understand how the function behaves instantly at any given point.

Graphing functions is a visual way to understand the behavior of functions over different intervals. It allows us to see patterns, trends, and points where the function behaves in a particular way. Let's consider our function \( f(x) = \frac{1}{x} - x^{2/3} \).

When graphing, look out for values where the function is not defined. For us, \( f(x) \) is undefined at \( x = 0 \) because \( \frac{1}{x} \) becomes problematic. This means the graph will have two separate parts: one for negative values and one for positive values.

Also, note the interaction with the tangent line \( y = -\frac{5}{3}x - \frac{11}{3} \). This line meets the graph at the point \( x = -1 \), \( y = -2 \). By plotting both the function and the tangent line together, you observe:

• The curve of the function touches the tangent line at this specific point.
• Elsewhere, the two separate lines never meet, highlighting how the derivative's information localizes the tangent exactly to its point of contact.

Such visual insight is invaluable to grasping how derivatives affect the shape and slope of graphs.

The power rule is a straightforward yet powerful tool for finding derivatives. It's especially useful when dealing with powers of \( x \) in functions. The rule states: if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). This simplifies derivative calculations enormously.

Using the power rule, we can find the derivative of \( x^{2/3} \). Applying \( f(x) = x^{2/3} \), the rule gives: \( f'(x) = \frac{2}{3}x^{-1/3} \). This transforms challenging algebra into a manageable form.

In our exercise, the power rule helped break down \( f(x) = \frac{1}{x} - x^{2/3} \):

• The term \( x^{2/3} \) yielded its derivative \( -\frac{2}{3}x^{-1/3} \).
• Coupled with \( \frac{1}{x} \)'s derivative of \( -\frac{1}{x^2} \), we derive \( f'(x) = -\frac{1}{x^2} - \frac{2}{3}x^{-1/3} \).

The power rule is pivotal for quickly computing derivatives, saving time and simplifying the mathematical process when dealing with polynomial expressions. Understanding and using this rule is a cornerstone of calculus skills.