SUMMARY
The derivative of the function G(x) = ∫ from x to x² of sin(-t²) dt is evaluated using the Fundamental Theorem of Calculus and the chain rule. The correct derivative is G'(x) = sin(-x⁴) * 2x - sin(-x²). This expression can be simplified to G'(x) = -sin(x²) + 2x * sin(x⁴). The discussion emphasizes the importance of correctly applying differentiation techniques to integral functions.
PREREQUISITES
- Understanding of the Fundamental Theorem of Calculus
- Proficiency in applying the chain rule in differentiation
- Familiarity with Taylor series expansions
- Knowledge of trigonometric identities, specifically sin(-θ) = -sin(θ)
NEXT STEPS
- Study the Fundamental Theorem of Calculus in detail
- Practice differentiation using the chain rule with various functions
- Explore Taylor series and their applications in calculus
- Review trigonometric identities and their implications in calculus problems
USEFUL FOR
Students preparing for calculus exams, particularly those focusing on derivatives of integral functions, as well as educators seeking to clarify concepts related to the Fundamental Theorem of Calculus and differentiation techniques.
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