Understanding the Derivative of Cosine Function

The video tutorial provides a step-by-step proof that the derivative of cosine x is negative sine x. It begins by defining the derivative using the limit of the difference quotient. The cosine of x plus h is expanded using the sum identity, and the expression is rewritten as a difference of two fractions. Factoring and special limits are applied to simplify the expression, leading to the conclusion that the derivative of cosine x is negative sine x. The tutorial concludes with a graph analysis showing the relationship between the cosine function and its derivative.

What is the derivative of cosine x according to the proof?

• Cosine x

• Sine x

• Negative cosine x

• Negative sine x

What is the difference quotient used in the proof?

• f(x) - f(x+h) / h

• f(x+h) - f(x) / h

• f(x) + f(x+h) / h

• f(x+h) + f(x) / h

Which identity is used to expand cosine of (x + h)?

• Sum identity

• Difference identity

• Quotient identity

• Product identity

What is the result of expanding cosine of (x + h) using the sum identity?

• cos(x) * cos(h) + sin(x) * sin(h)

• cos(x) + cos(h) + sin(x) + sin(h)

• cos(x) * cos(h) - sin(x) * sin(h)

• cos(x) - cos(h) - sin(x) - sin(h)

How is the expression rewritten as a difference of two limits?

• By subtracting two fractions

• By dividing two fractions

• By multiplying two fractions

• By adding two fractions

What common factor is factored out from the first limit?

• Sine x

• Cosine x

• Negative cosine x

• Negative sine x

What is the value of the limit as h approaches zero for sine(h)/h?

• 0

• 1

• Undefined

• Infinity

What is the final simplified result of the proof?

• Sine x

• Negative sine x

• Cosine x

• Negative cosine x

What does the graph of the derivative function represent?

• The integral of the original function

• The area under the original function

• The original function

• The slopes of tangent lines to the original function

Where do the slopes of the tangent lines to the cosine function equal zero?

• At pi/3 and 2pi/3

• At pi/4 and 3pi/4

• At 0, pi, and 2pi

• At pi/2 and 3pi/2