Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
y = x⁵

Verified step by step guidance

1. Step 1: Identify the function for which you need to find the derivative. Here, the function is \( y = x^5 \).
2. Step 2: Recognize that this is a power function of the form \( y = x^n \), where \( n = 5 \).
3. Step 3: Apply the power rule for differentiation, which states that the derivative of \( x^n \) is \( nx^{n-1} \).
4. Step 4: Substitute \( n = 5 \) into the power rule formula to find the derivative: \( \frac{d}{dx}(x^5) = 5x^{5-1} \).
5. Step 5: Simplify the expression to obtain the derivative: \( 5x^4 \).

Key Concepts

Derivatives

A derivative represents the rate at which a function changes at a given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the graph of the function at a specific point.

Power Rule

The Power Rule is a basic differentiation rule used to find the derivative of functions in the form of x^n, where n is a real number. According to this rule, the derivative of x^n is n*x^(n-1). This rule simplifies the process of differentiation for polynomial functions, making it easier to compute derivatives quickly.

Function Notation

Function notation is a way to represent functions and their outputs. In the context of derivatives, it is common to express a function as y = f(x), where f(x) denotes the function's output for a given input x. Understanding function notation is essential for applying differentiation techniques and interpreting the results correctly.