Understanding Derivatives of Trigonometric Functions

The video presents a visual argument to demonstrate why the derivative of cosine of x is negative sine of x. It builds on a previous proof that the derivative of sine of x is cosine of x. By shifting the graphs of sine and cosine to the left by pi/2, the video shows that the derivative relationship holds, using trigonometric identities to validate the argument.

What is the main goal of the visual argument presented in the video?

• To show the derivative of cosine is negative sine

• To explain the concept of limits

• To prove the derivative of sine is cosine

• To demonstrate the addition of angles in trigonometry

What previous knowledge is assumed in the video?

• Derivative of sine is cosine

• Derivative of cosine is sine

• Derivative of tangent is secant squared

• Derivative of secant is tangent

What does the blue graph represent in the context of the video?

• The original function

• The derivative of the red graph

• The integral of the red graph

• A random trigonometric function

What transformation is applied to both graphs in the video?

• Shift to the left by pi/2

• Shift to the right by pi/2

• Vertical stretch

• Horizontal compression

What is the purpose of shifting the graphs by pi/2?

• To change the amplitude

• To demonstrate a phase shift

• To prove a trigonometric identity

• To simplify the function

After shifting, what does the red graph represent?

• Sine of x + pi/2

• Sine of x

• Cosine of x + pi/2

• Cosine of x

According to trigonometric identities, what is sine(x + pi/2) equivalent to?

• Negative cosine of x

• Sine of x

• Negative sine of x

• Cosine of x

What is the equivalent expression for cosine(x + pi/2) using trigonometric identities?

• Negative sine of x

• Sine of x

• Negative cosine of x

• Cosine of x

What is the relationship between the red and blue graphs after the shift?

• Both graphs are identical

• The blue graph is the derivative of the red graph

• The graphs are unrelated

• The red graph is the derivative of the blue graph

What is the conclusion of the visual argument regarding the derivative of cosine?

• It is equal to sine

• It is equal to cosine

• It is equal to negative cosine

• It is equal to negative sine