Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(x) = 5x³

Verified step by step guidance

1. Step 1: Identify the function for which you need to find the derivative. Here, the function is \( f(x) = 5x^3 \).
2. Step 2: Recall the power rule for differentiation, which states that if \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \).
3. Step 3: Apply the power rule to the function \( f(x) = 5x^3 \). Here, \( a = 5 \) and \( n = 3 \).
4. Step 4: Differentiate the function using the power rule: \( f'(x) = 5 \times 3x^{3-1} \).
5. Step 5: Simplify the expression obtained from differentiation: \( f'(x) = 15x^2 \).

Key Concepts

Derivatives

A derivative represents the rate at which a function changes at any given point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In practical terms, the derivative provides the slope of the tangent line to the curve of the function at a specific point.

Power Rule

The Power Rule is a fundamental technique for finding derivatives of polynomial functions. It states that if f(x) = x^n, where n is a real number, then the derivative f'(x) = n*x^(n-1). This rule simplifies the process of differentiation for functions involving powers of x.

Function Notation

Function notation is a way to represent mathematical functions in a clear and concise manner. In this context, f(x) denotes a function of x, allowing us to express the relationship between the input x and the output f(x). Understanding function notation is essential for applying calculus concepts, including differentiation.