What is the derivative of with respect to ?
Answer and explanation
Correct answer: B. applying the chain rule with FTC
- Step 1: By the Leibniz integral rule with chain rule: \frac{d}{dx} \left[ \int_0^{u(x)} f(t) \, dt ight] = f(u(x)) \cdot u'(x). Here , and . Therefore . **Answer: applying the chain rule with FTC. Why not the others:** " missing chain factor" omits the derivative of the upper limit ; " un-squared exponent" fails to square the substituted upper limit, writing instead of ; " naive substitution" substitutes directly without accounting for the limit or chain rule.
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