Calculus Derivatives and Limits

This video tutorial focuses on the derivatives of sine and cosine functions. It begins with an introduction to the basic derivatives of sine and cosine, followed by example problems using the constant multiple rule. The tutorial then progresses to more advanced examples involving constants. It also demonstrates how to prove the derivative of sine using the limit definition and explains special trigonometric limits necessary for understanding these derivatives.

What is the derivative of sine x?

• cosine x

• negative sine x

• secant x

• tangent x

What is the derivative of cosine x?

• negative sine x

• sine x

• negative cosine x

• tangent x

Using the constant multiple rule, what is the derivative of 4 sine x?

• negative 4 cosine x

• negative 4 sine x

• 4 sine x

• 4 cosine x

What is the derivative of 8 cosine x using the constant multiple rule?

• 8 cosine x

• negative 8 cosine x

• negative 8 sine x

• 8 sine x

What is the derivative of 3 cosine x minus pi sine x?

• negative 3 sine x minus pi cosine x

• negative 3 cosine x plus pi sine x

• 3 sine x minus pi cosine x

• 3 cosine x plus pi sine x

How can sine x divided by 5 be rewritten to apply the constant multiple rule?

• one-fifth times sine x

• sine x divided by 5

• sine x times 5

• 5 times sine x

What is the limit definition of a derivative?

• f prime of x equals the limit as h approaches zero of f of x plus h minus f of x divided by h

• f prime of x equals the limit as h approaches zero of f of x times h

• f prime of x equals f of x plus h minus f of x

• f prime of x equals f of x times h

What is the limit as x approaches zero for sine x over x?

• undefined

• infinity

• 1

• 0

What is the limit as x approaches zero for one minus cosine x over x?

• 0

• 1

• infinity

• undefined

What is the final result of the derivative of sine x using the limit definition?

• sine x

• negative sine x

• cosine x

• negative cosine x