Find the derivative of f(x)=xsin(x)

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BlackTom AI · Accepted Answer

Rewriting the problem in my own words: Find the derivative of f(x) = x sin(x).

First, I note that this function is a product of two functions, x and sin(x). To differentiate a product, I should apply the product rule: (u v)' = u' v + u v'. Here, let u = x and v = sin(x). Then u' = 1 and v' = cos(x).

Applying the product rule gives f'(x) = u' v + u v' = (1)(sin(x)) + (x)(cos(x)) = sin(x) + x cos(x).

If the answer choices (were any) reflected this result, they would show f'(x) = x cos(x) + sin(x), which is algebraically equivalent to sin(x) + x cos(x). The exact order of terms does not affect equivalence since addition is commutative.

Common pitfalls to avoid:
- Forgetting the product rule by treating x sin(x) as a simple single function and differentiating only one part.
- Misidentifying v' as cos(x) without applying the product rule properly.
- Incorrectly differentiating sin(x) as cos(x) without multiplying by x when applying the product rule.

Summary: The derivative of f(x) = x sin(x) is sin(x) + x cos(x), which can also be written as x cos(x) + sin(x).

DIFFERENTIAL CALCULUS (MTH_251_020_W2025) Unit 3 - Trig and Exponential - PreClass

Find the derivative of f(x)=xsin(x)

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