Intro Calc – Difference Quotient and Power Rule – Discovering Derivatives
Name:_______________
𝑓(𝑥+ℎ)−𝑓(𝑥)
)
ℎ
1) Using the definition of the difference quotient determine the lim (
ℎ→0
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
for the function f(x) = 4x² - 6x +2
8x - 6
What does the answer you obtained in number 1 tell you __________________
The slope at any point for f(x) / The derivative of f(x)
Determine the slope of the tangent line at x = 1 for the function f(x) in problem number 1.
The slope is 2
At which value of x is the slope of the tangent line to the equation in number 1 equal to zero?
x = 3/4
At which value of x is the slope of the tangent line to the equation in number 1 equal to negative
two?
x = 1/2
Show the difference quotient proof for the function f(x)= √x
See notes from class.
Use the power rule to differentiate f(x) = 7x³ - 10x² + 2x -9
f’(x) = 21x² - 10x + 2
If you were to use the difference quotient in problem 7, what would your answer be when you let
the limit of h be 0?
f’(x) = 21x² - 10x + 2
For the equation in number 7, what is the slope at the origin?
2
For the equation in number 7, what is the slope of the tangent line at x = 1?
13
What it the derivative of a constant? Explain. (Conceptually speaking, why does this make sense?)
The derivative of a constant is zero. Conceptually speaking, this makes sense because the derivative
allows you to calculate the slope of a function at any point. A constant function, for example y = c,
will have a graph which is a horizontal line intersecting the y-axis at c. The slope of a horizontal line
is always 0. Thus, the derivative of a constant will always be zero.
12) What is the derivative of f(x) = 3x? Prove this.
f’(x) = 3 ----- You may provide a proof using a difference quotient and taking the limit as h
approaches zero, or you may simply justify your answer using the power rule and differentiating f(x)
= 3x showing all steps.