Question

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

Short answer

\(2x\)

Step-by-step solution

Identify the function to differentiate

First, recognize the function you need to differentiate, which is a polynomial function: \(f(x) = x^{2} - 1\).

Apply the power rule

For each term of the polynomial, apply the power rule of differentiation, which says that if \(f(x) = x^n\), then \(f'(x) = n \times x^{(n-1)}\).

Differentiate term by term

Differentiate the first term \(x^{2}\) using the power rule to get \(2x\), and differentiate the constant term \(-1\) to get 0 (since the derivative of any constant is 0).

Combine the derivatives

Combine the derivatives of the individual terms to obtain the derivative of the entire function: \(f'(x) = 2x + 0 = 2x\).

Key concepts

Power Rule Differentiation

Power rule differentiation is a basic yet extremely powerful tool in calculus, especially when dealing with polynomial functions. The rule is straightforward: for a given function in the form of \(f(x) = x^n\), where \(n\) is any real number, the derivative of this function with respect to \(x\) is found by multiplying \(n\) with the power of \(x\) to the \((n-1)\)-th power. In other words, \(f'(x) = n \times x^{(n-1)}\).

For example, if you have a function like \(g(x) = x^5\), applying the power rule yields \(g'(x) = 5x^4\). It’s essential to note that this rule applies to each term independently within a polynomial, which leads to its effectiveness in term-by-term differentiation. The power rule becomes instrumental in simplifying the process of finding derivatives, making it a fundamental skill for students to master in calculus.

Polynomial Function Differentiation

When differentiating polynomial functions, which are functions that consist of terms with variables raised to whole number powers and possibly including constant terms, a methodical approach is taken. Each term in the polynomial is differentiated individually, and the result is the combination of the derivatives of those terms.

Consider a function like \(h(x) = 4x^3 - 3x^2 + 2x - 5\). You would differentiate each term separately: the \(4x^3\) term becomes \(12x^2\), the \(-3x^2\) term becomes \(-6x\), the \(2x\) term becomes 2, and the constant \(-5\) becomes 0. The derivative of the polynomial function is then \(h'(x) = 12x^2 - 6x + 2\). Polynomial function differentiation highlights the power rule's versatility as it's applied term by term to successfully compute the derivative of an entire polynomial expression.

Differentiate Term by Term

The process of differentiating a polynomial term by term is akin to simplifying a complex problem into smaller, more manageable parts. When a polynomial consists of multiple terms, each term is treated as a separate function for differentiation purposes. By applying the power rule to each term individually and then summing the results, we find the derivative of the entire polynomial.

For instance, let's differentiate \(j(x) = x^4 - 6x^2 + 3\) term by term. We differentiate \(x^4\) to get \(4x^3\), \(-6x^2\) to get \(-12x\), and the derivative of the constant 3 is 0. Therefore, \(j'(x) = 4x^3 - 12x\). This method ensures that each term receives the proper treatment according to the power rule, resulting in an accurate overall derivative. It allows for a systematic approach to handling more complex functions, ensuring that the differentiation process is thorough and clear.