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The power rule is a fundamental concept in calculus. It makes finding the derivative of power functions very straightforward. Specifically, the power rule states:

• If your function is in the form of f(x) = axⁿ, then its derivative is f'(x) = anx^n-1.

Here:

• a is a constant coefficient
• n is the exponent

In our problem, the function is f(x) = -x².

• In this case, a is -1 and n is 2.

Using the power rule formula, finding the derivative of a power function becomes as simple as plugging in the values for a and n.

A derivative represents the rate at which a function is changing at any given point. Conceptually, it's the slope of the tangent line to the function's graph at that point.

In simpler terms, it tells us how fast the function's value is increasing or decreasing. For instance, if the function measures distance over time, the derivative measures velocity.

In symbolic terms, if you have a function f(x), its derivative is denoted as f'(x) or sometimes df/dx. This new function, the derivative, can be used to understand the behavior of the original function.

• If f'(x) is positive, the function is increasing at that point.
• If f'(x) is negative, the function is decreasing at that point.
• If f'(x) is zero, the function has a stationary point (which could be a maximum, minimum, or inflection point).

To differentiate a function using the power rule, you typically follow a set of steps. Let's apply these steps to the given function f(x) = -x²:

Step 1: Rewrite the Function

If needed, rewrite the function in a more convenient form for differentiation. In this case, the function is already simple enough: f(x) = -x².

Step 2: Identify the Components

Identify the coefficient (a) and the exponent (n) in your function. Here, a is -1 and n is 2.

Step 3: Apply the Power Rule

Now, use the power rule formula: \[ f'(x) = anx^{n-1} \]
For our function: \[ f'(x) = (-1) \times 2 \times x^{2-1} = -2x \]
This gives us the derivative of the function, which is -2x.

• Always remember to multiply the coefficient by the exponent.
• Then, reduce the exponent by one.

And there you have it! The derivative of f(x) = -x² is f'(x) = -2x. Understanding this process will make it easier for you to tackle more complex differentiation problems in the future.