7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
f(w) = w³ -w / w

검증된 단계별 안내

Identify the function f(w) = \(\frac{w^3 - w}{w}\) and simplify it by dividing each term in the numerator by the denominator.

Simplify the expression: \(\frac{w^3}{w}\) - \(\frac{w}{w}\) = w^2 - 1.

Recognize that the simplified function f(w) = w^2 - 1 is a polynomial function.

Differentiate the simplified function using the power rule: \(\frac{d}{dw}\)(w^2) = 2w and \(\frac{d}{dw}\)(-1) = 0.

Combine the derivatives to find the derivative of the function: f'(w) = 2w.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Product Rule

The Product Rule is a formula used to find the derivative of the product of two functions. If you have two functions, u(w) and v(w), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating expressions where two functions are multiplied together, allowing for a systematic approach to finding the derivative.

Quotient Rule

The Quotient Rule is used to differentiate functions that are expressed as the ratio of two other functions. If f(w) = u(w)/v(w), the derivative is given by (u'v - uv')/v². This rule is particularly useful when dealing with fractions in calculus, ensuring that the differentiation accounts for both the numerator and denominator.

Simplification of Derivatives

Simplification of derivatives involves reducing the expression obtained after applying differentiation rules to its simplest form. This may include factoring, canceling common terms, or combining like terms. Simplifying the result is crucial for clarity and ease of interpretation, especially when further analysis or evaluation is required.

7–14. Find the derivative the following ways:a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.f(w) = w³ -w / w