91Ó°ÊÓ

A derivative is a fundamental concept in calculus. It helps us understand how a function changes as its input changes. When we find the derivative of a function, we're essentially calculating the rate of change or the "instantaneous speed" of the function's output with respect to its input.

To find the derivative of a function, we frequently use differentiation rules. This allows us to perform calculus operations to identify how each part of the function contributes to changes. For the given function, \[ f(x) = 3x^2 - x^3, \]we differentiate each term separately using these rules. The differentiation provides\[ f'(x) = 6x - 3x^2, \]that represents the rate of change of the original function. Using these derivative results, you can then answer questions about the behavior of the function at any given point.

The slope of a tangent line at a specific point on a curve gives us an exact value of how steep the tangent is. It tells how quickly or slowly the function is increasing or decreasing at that particular location.

To find the slope of the tangent line, we use the derivative. Once we have the derivative, we substitute the x-coordinate of the point of interest into this derivative equation. In our exercise, this means inserting \(x=1\) into\[ f'(x) = 6x - 3x^2 \] to find that the slope at (1, 2) is\[ f'(1) = 3. \]

• Derivatives give the slope.
• We compute them at specific points for precise information.
• Tangent lines tell us about local behavior of functions.

Differentiation rules are the tools we use to calculate derivatives efficiently. These rules simplify the process of finding how different parts of a function behave. Here are some basic rules we often use:

• Power Rule: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
• Constant Multiple Rule: If \( f(x) = c \, g(x) \), then \( f'(x) = c \, g'(x) \), where \(c\) is a constant.
• Sum Rule: If \( f(x) = g(x) + h(x) \), then \( f'(x) = g'(x) + h'(x) \).

In our example, we applied the power rule and the constant multiple rule to differentiate\[ f(x) = 3x^2 - x^3. \]
Using these rules effectively gives\[ f'(x) = 6x - 3x^2, \]enabling us to find the necessary slope and equation of the tangent line. The rules provide a structured path to understanding and solving differentiation problems.