Understanding Derivatives in Calculus

The video tutorial explains the derivatives of sine and cosine functions, focusing on proving the derivative of sine using limits. It introduces the concept of derivatives, applies trigonometric identities, and performs limit calculations to demonstrate the proof. The tutorial concludes by highlighting key limit results essential for understanding calculus.

What is the derivative of sine of x with respect to x?

• Negative cosine of x

• Sine of x

• Cosine of x

• Negative sine of x

What is the derivative of cosine of x with respect to x?

• Negative cosine of x

• Sine of x

• Cosine of x

• Negative sine of x

What is the limit definition of the derivative of sine of x?

• Limit as delta x approaches 0 of (cosine of x plus delta x minus cosine of x) over delta x

• Limit as delta x approaches 0 of (sine of x plus delta x minus sine of x) over delta x

• Limit as delta x approaches 0 of (sine of x plus delta x plus sine of x) over delta x

• Limit as delta x approaches 0 of (cosine of x plus delta x plus cosine of x) over delta x

Which trigonometric identity is used to rewrite sine of x plus delta x?

• Cosine of x times sine of delta x plus sine of x times cosine of delta x

• Sine of x times sine of delta x plus cosine of x times cosine of delta x

• Sine of x times cosine of delta x minus cosine of x times sine of delta x

• Cosine of x times cosine of delta x minus sine of x times sine of delta x

What is the result of the limit as delta x approaches 0 of sine of delta x over delta x?

• 0

• 1

• Infinity

• Undefined

What is the result of the limit as delta x approaches 0 of (1 minus cosine of delta x) over delta x?

• Undefined

• 0

• 1

• Infinity

What theorem is used to prove the limit of sine of delta x over delta x?

• Pythagorean Theorem

• Squeeze Theorem

• Fundamental Theorem of Calculus

• Mean Value Theorem

What is the final result of the derivative of sine of x?

• Cosine of x

• Negative sine of x

• Negative cosine of x

• Sine of x

What happens to the term involving sine of x times 0 in the final expression?

• It becomes infinity

• It remains unchanged

• It becomes 0

• It becomes 1

Why is it important to understand the derivative of sine and cosine?

• They help in solving complex derivatives

• They are the only derivatives needed in calculus

• They are only used in trigonometry

• They are not important