By computing the first few derivatives and looking for a pattern, find the following derivatives.

c. d⁷³/dx⁷³ (x sin x)

검증된 단계별 안내

First, identify the function for which you need to find the 73rd derivative: f(x) = x sin(x).

Compute the first derivative using the product rule: d/dx [x sin(x)] = x d/dx[sin(x)] + sin(x) d/dx[x].

The first derivative is: f'(x) = x cos(x) + sin(x).

Compute the second derivative by applying the product rule to each term in f'(x): f''(x) = d/dx[x cos(x)] + d/dx[sin(x)].

Continue finding higher-order derivatives and look for a pattern in the derivatives to generalize the 73rd derivative.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

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

Product Rule

The product rule is a fundamental technique in calculus used to find the derivative of a product of two functions. If you have two differentiable functions, u(x) and v(x), the derivative of their product is given by (uv)' = u'v + uv'. This rule is essential when differentiating expressions like x sin(x), where both x and sin(x) are functions of x.

Higher Order Derivatives

Higher order derivatives refer to the derivatives of a function taken multiple times. For example, the second derivative is the derivative of the derivative, and so on. In this problem, you are asked to find the 73rd derivative, which involves repeatedly applying differentiation rules and identifying any emerging patterns in the derivatives.

Pattern Recognition in Derivatives

When computing higher order derivatives, recognizing patterns can simplify the process. For functions like x sin(x), derivatives often follow a cyclical pattern due to the periodic nature of trigonometric functions. Identifying these patterns allows for efficient computation of derivatives without performing each differentiation step manually.

By computing the first few derivatives and looking for a pattern, find the following derivatives.c. d⁷³/dx⁷³ (x sin x)