2.5 Applying the Power Rule

The Power Rule is a fundamental concept in Calculus that allows us to differentiate functions that have a power term. It states that the derivative of x^n, where n is a constant, is nx^(n-1). In this guide, we will be discussing how to apply the Power Rule to different types of functions and how it can be used to simplify the process of finding derivatives. The Power Rule states that the derivative of x^n is nx^(n-1). This means that if we have a function of the form f(x) = x^n, we can find its derivative by simply multiplying n by x^(n-1). For example, the derivative of x^2 is 2x^(2-1) = 2x^1 = 2x. It is important to note that the Power Rule only applies when the exponent is a constant. If the exponent is a variable or a function, we cannot use the Power Rule and must use other methods such as chain rule to find the derivative. The Power Rule can also be applied to functions that are multiplied by a constant. For example, if we have a function of the form f(x) = 3x^2, the derivative would be 3*2x^(2-1) = 6x. The constant is simply distributed to the exponent. Another application of the Power Rule is finding the derivatives of inverse functions. For example, the inverse of f(x) = x^2 is f^-1(x) = √x. To find the derivative of f^-1(x), we can apply the Power Rule to the original function f(x) = x^2 and then use the chain rule. The derivative of x^2 is 2x, so the derivative of f^-1(x) = √x is (1/2)x^(-1/2). In addition to these examples, the Power Rule can also be applied to more complex functions by breaking them down into simpler terms. For example, if we have a function f(x) = (3x^2 + 4x - 2)^5, we can use the Power Rule to find the derivative by applying the rule to each term individually and then applying the product rule.

Example Problems: Example 1: Find the derivative of f(x) = x^4 Solution: Using the Power Rule, we know that the derivative of x^n is nx^(n-1). In this case, n = 4, so the derivative of f(x) = x^4 is 4x^(4-1) = 4x^3. Example 2: Find the derivative of f(x) = 5x^2 Solution: The derivative of f(x) = 5x^2 is 5*2x^(2-1) = 10x. The constant 5 is simply distributed to the exponent. Example 3: Find the derivative of f^-1(x) = √x Solution: Using the Power Rule, we know that the derivative of x^2 is 2x. Since f(x) = x^2 is the inverse of f^-1(x) = √x, we can use the chain rule to find the derivative of f^-1(x). The derivative of f^-1(x) = (1/2)x^(-1/2). Applying the Power Rule is a fundamental skill in Calculus and can greatly simplify the process of finding derivatives. By understanding how to apply the Power Rule to different types of functions and when it can be used, we can quickly and easily find the derivative of many functions. Remember to always check if the exponent is a constant before applying the Power Rule and to use other methods such as chain rule when the exponent is a variable or function. Practice different types of problems to solidify your understanding of the Power Rule and its applications.