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One of the fundamental techniques in calculus is finding the derivative of a function, which provides us the rate at which the function's value changes with respect to a change in its input value or variable. The power rule is a quick tool for differentiating functions of the form

\( x^n \), where \(n\) is any real number. According to this rule, the derivative of \(x^n\) is \(nx^{n-1}\). So, for any term in a polynomial where the variable \(x\) is raised to a power, you simply multiply the term by the exponent and then subtract one from the exponent.

For example, in the exercise \(f(x)=x^{2}-9\), applying the power rule to the term \(x^2\) gives us \(2x^{2-1}\) or \(2x\), as the derivative with respect to \(x\). The constant term \(-9\) drops out because the derivative of any constant is zero. This powerful and straightforward concept allows quick derivatives of polynomials without complex calculations.

Polynomials are expressions that are composed of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

To find the derivatives of polynomials, we utilize the fact that the derivative of a sum of terms is the sum of the derivatives of those terms. This is an application of the sum rule in differentiation. For our example \(f(x)=x^{2}-9\), we take the derivative of each term separately. The first term \(x^{2}\) differentiates to \(2x\), using the power rule, and for any constant such as -9, the derivative is 0, as constants do not change and hence their instantaneous rate of change is zero.

The end result is a polynomial where each term has been differentiated separately. This is particularly useful since it breaks down complex expressions into simpler parts that can be handled individually.

Once we have the derivative of a function, evaluating it at specific points gives us the slope of the tangent line to the function's graph at those points. This slope indicates how steep the graph is at that point, and the direction it is moving, either upwards or downwards.

To evaluate derivatives, we simply substitute the desired \(x\)-values into the derivative formula. In the given exercise, after finding the derivative \(f'(x)=2x\), we evaluate it at \(x=2\) to obtain \(f'(2)=2*2=4\) and at \(x=-2\) to obtain \(f'(-2)=2*(-2)=-4\). These calculated values indicate that the slope of the original function \(f(x)=x^{2}-9\) at \(x=2\) is 4, which means the graph is rising, and at \(x=-2\) it's -4, indicating the graph is falling at this point.

Evaluating derivatives at particular points is particularly useful in many applications of calculus, including optimization problems where you need to find maximum or minimum values of functions.