Question

Find the derivative. $$D\left(x^{2}-1\right)$$

Short answer

\(D(x^{2}-1) = 2x\)

Step-by-step solution

Identify the function to differentiate

We have the function of a single variable given by \(f(x) = x^{2} - 1\). The task is to find the derivative of this function with respect to \(x\), which is denoted by \(D(f(x))\) or \(f'(x)\).

Apply the power rule of differentiation

To differentiate \(x^{2}\), use the power rule: \(D(x^n) = nx^{n-1}\). Applying this to \(x^{2}\) we have \(D(x^{2}) = 2x^{2-1} = 2x\). The derivative of a constant, such as -1, is 0. Therefore, \(D(-1) = 0\).

Combine the derivatives

Combine the derivatives of the individual terms. The derivative of the function \(f(x) = x^{2} - 1\) with respect to \(x\) is the derivative of \(x^{2}\) minus the derivative of \(-1\). Thus, we have \(f'(x) = D(x^{2}) - D(-1) = 2x - 0 = 2x\).

Key concepts

Power Rule Differentiation

Understanding the power rule for differentiation is crucial in calculus education. This rule simplifies the process of finding derivatives, which are fundamental in analyzing the rate of change in functions. When you differentiate a term like x^n, where n is any real number, you simply bring down the exponent to multiply in front of the variable, and then subtract one from the exponent.
In mathematical terms, the power rule states that if you have a function f(x) = x^n, its derivative f'(x) is given by f'(x) = n*x^{n-1}. This is precisely what was applied in the textbook solution, where the function x^2 was differentiated using the power rule to get 2x.
For calculus students, it's worth noting that this rule only applies to functions where the power of x is a constant. This makes the power rule a quick and efficient method to calculate derivatives, especially for polynomial functions.

Derivative of a Function

The derivative of a function represents the slope of the curve at any given point on a graph of the function. It's an essential concept in both physics and mathematics as it describes how one variable changes in relation to another. In the context of our textbook solution, finding the derivative of f(x) = x^2 - 1 involves understanding that the derivative signifies the instantaneous rate of change of the function.

Breaking Down the Derivative

For a function f(x), the derivative f'(x) or D(f(x)) is the limit of the average rate of change as the interval approaches zero. Calculating the derivative using basic rules such as the power rule, sum rule, and constant multiple rule, as demonstrated in our exercise, makes this process much more manageable. It is a compilation of several simpler operations rather than a single complex calculation.

Calculus Education

Calculus is the mathematical study of continuous change, and it is a vital subject for students in various scientific and engineering fields. Effective calculus education focuses on helping students understand the underlying concepts of limits, derivatives, integrals, and the fundamental theorem of calculus.
For beginners, starting with the foundational rules—like the power rule for differentiation—is crucial in building a strong understanding before moving to more complex aspects of calculus such as integral calculus or multivariable calculus.
An educator should emphasize visualization by using graphs and real-life applications to show the practicality of calculus. Creating a narrative or a story for problems, like how changing conditions affect the trajectory of a projectile, can make calculus more relatable and interesting, enhancing students' comprehension and retention of complex mathematical concepts.