Understanding the Derivative of Sine X

The video tutorial provides a detailed proof that the derivative of sine X with respect to X equals cosine X. It begins by introducing the definition of the derivative and applies it to sine X. The tutorial then expands sine of X plus H using the sum identity, factors and simplifies the expression, and uses special limits to conclude the proof. Finally, it offers a graphical interpretation of sine and cosine functions, highlighting how the cosine function values represent the slopes of tangent lines to the sine function.

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

• Sine X

• Cosine X

• Tangent X

• Secant X

Which identity is used to expand sine(X + h)?

• Quotient identity

• Sum identity

• Difference identity

• Product identity

What common factor is factored out from the terms in the numerator during simplification?

• Cosine X

• Sine X

• Tangent X

• Secant X

What is the limit of sine H divided by H as H approaches zero?

• Infinity

• 1

• Undefined

• 0

What is the limit of (1 - cosine H) divided by H as H approaches zero?

• 1

• 0

• Infinity

• Undefined

What does the cosine function represent in the context of the derivative of sine X?

• The rate of change of sine X

• The maximum value of sine X

• The integral of sine X

• The area under sine X

What is the significance of the points where the derivative of sine X is zero?

• They are points of maximum curvature

• They are points where the function is undefined

• They are points of inflection

• They are points where the tangent is horizontal

How are the graphs of sine X and its derivative related?

• The graph of the derivative is a vertical shift of sine X

• The graph of the derivative is a horizontal shift of sine X

• The graph of the derivative is a reflection of sine X

• The graph of the derivative is a phase shift of sine X

What does a horizontal tangent line on the graph of sine X indicate?

• A point of intersection with the x-axis

• A maximum or minimum point

• A point of inflection

• A point of discontinuity

What is the relationship between the slopes of tangent lines to sine X and the cosine function?

• The slopes are equal to the cosine function

• The slopes are equal to the secant function

• The slopes are equal to the sine function

• The slopes are equal to the tangent function