Understanding Derivatives and Their Properties

The video tutorial explains limit-based definitions of derivatives, focusing on the derivative of a constant function. It demonstrates why the derivative of a constant is zero using graphical and limit-based approaches. The tutorial also provides an example to reinforce the concept.

What are the two main perspectives of limit-based definitions of derivatives discussed?

• Derivative of a polynomial and a rational function

• Derivative of a function and its inverse

• Derivative of a constant and a variable

• Derivative at a point and derivative in general

What is the derivative of a constant function?

• Equal to the constant

• Equal to zero

• Undefined

• Equal to one

Why does the derivative of a constant function make intuitive sense?

• Because the function has a maximum point

• Because the function is always decreasing

• Because the function's value does not change

• Because the function is always increasing

In the graph of a constant function, what is the slope of the tangent line?

• Positive

• Negative

• Zero

• Undefined

What does the graph of a constant function look like?

• A diagonal line

• A parabolic curve

• A vertical line

• A horizontal line

How is the derivative of a constant function proven using limits?

• By showing the limit approaches infinity

• By showing the limit approaches zero

• By showing the limit does not exist

• By showing the limit approaches the constant

What happens to the limit of (f(x + h) - f(x))/h as h approaches zero for a constant function?

• It approaches infinity

• It becomes undefined

• It approaches the constant

• It approaches zero

If h(x) = Pi, what is the derivative h'(x)?

• Pi

• Zero

• Undefined

• One

Why is the derivative of h(x) = Pi equal to zero?

• Because Pi is undefined

• Because Pi is a function

• Because Pi is a constant

• Because Pi is a variable

What does the instantaneous rate of change of a constant function indicate?

• The function is constant

• The function is oscillating

• The function is decreasing

• The function is increasing