DEFINITION OF DERIVATIVE
What does a derivative represent?
Instantaneous rate of change
What is the difference quotient
(f(x+h)−f(x))/h
If a function is differentiable, it must be what?
Continuous
Derivative of a constant
0
What does a tangent line touch?
One point on the curve
What geometric meaning does the derivative have?
Slope of the tangent line
What does the difference quotient measure?
Average rate of change
Can a function be continuous but not differentiable?
Yes
Derivative of (x)^2
2x
What does slope = 0 mean?
Horizontal tangent
What is f′(x)?
A function giving slope at every x
What happens to h in the derivative definition?
h → 0
Example of non-differentiable function?
∣x∣ at x = 0
Power rule
nx^(n-1)
If derivative is positive, function is
Increasing
What does “instantaneous” mean in calculus?
At a single point
What type of line does the difference quotient represent?
Secant line
Name one place derivative does not exist
Corner / cusp / discontinuity / vertical tangent
Derivative of 3x^(2) + 4x
6x + 4
If derivative is negative, function is
Decreasing
What real-world concept is similar to a derivative?
Speed (velocity)
What does taking the limit turn the secant line into?
Tangent line
Why is a function not differentiable at a corner?
Left and right slopes are different
Derivative of 5x^(4) − 2x^(2) + 7
20x^(3) - 4x
Find slope of f(x)=x^(2) at x = 3
6