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MATH 1170 Section 3.1 Worksheet
NAME
Recall that we have defined the derivative of f to be
f โฒ(x) = lim hโ 0
f (x + h) โ f (x) h
Recall also that the derivative of a function, f โฒ(x), is the slope f โs tangent line at the point (x, f (x)).
We are now going to explore what generalizations we can make about the certain kind of functions we have previously discussed.
Letโs begin with polynomial functions. The most basic polynomial function is the constant function.
Constant Functions
Recall that a constant function is a function f (x) = c where c is a constant (real number).
Sketch a graph of the function f (x) = c.
What is the slope of this line?
Looking at the graph, what would you imagine that the slope a line tangent to f (x) = c at any given is?
Using the definition of derivative, compute the derivative of f (x) = 3.
Using the definition if derivative, compute the derivative of f (x) = c.
Derivative of the Constant Function
d dx
(c) =
This little rule that we established makes it easy to compute derivatives of functions like:
f (x) = โ 90 g(t) = 17, 562 h(r) = ฯ^2 / 4 k(x) = eโ^1000 s(t) = โ
...they are all...0!
Power Functions
Sketch a graph of f (x) = x.
Using the definition of the derivative, compute the derivative of f (x) = x.
Using the definition of derivative, compute the derivative of f (x) = x^2.
Using the definition of derivative, compute the derivative of f (x) = x^3.
Exponential Functions
Recall that and exponential function comes in the form of f (x) = ax^ where a is some constant. Note that these functions are very different from power functions that have the form f (x) = xa^ where a is a constant.
Sketch the graphs of f (x) = 2x^ and f (x) = x^2.
Using the definition of derivative, compute the derivative of f (x) = ax.
Compare lim hโ 0
ak^ โ 1 h to the definition of the derivative of f (x) = ax^ at 0. What do you notice?
The Derivative of the Exponential Function
d dx
(ax) =
The definition of e is such that
lim hโ 0
eh^ โ 1 h
What does this tell you about the derivative of ex?
The Derivative of the Natural Exponential Function
d dx (ex) =
Composition of Derivatives
Now that we have established a few basic rules for derivatives, it would be beneficial to figure out how to combine them. For instance, we know that the derivative of f (x) = x^5 ,
d dx (x^5 ) = 5x^5 โ^1 = 5x^4.
But what about g(x) = โ 2 x^5?
Using the definition of derivative, compute the derivative of f (x) = 2x.(โ)
Using the definition if derivative, compute the derivative of f (x) = โ 3 x^2 .(โ)
Using the definition of derivative, compute the derivative of g(x) = cf (x) where c is a constant and f is a differentiable function.
Hint: Refer to the rules of limits on the review sheet.
The Sum Rule
If f and g are differentiable functions, then
d dx
(f (x) + g(x)) =
Using the Sum Rule, compute the derivative of f (x) = x^2 + x.
Can you make a conjecture about the derivative a the difference of two differentiable functions? What do
you think that (^) dxd (f (x) โ g(x)) will be and why?
The Difference Rule
If f and g are differentiable functions, then d dx (f (x) โ g(x)) =
Putting it Together
Differentiate the following functions using the rules that you established.
a. f (x) = x^144 โ 11 ex^ b. f (x) = 3x^4 (
x^3 + 15) c. f (x) = 4x^2 โ 2 x + 1 + 3xโ^1
d. f (x) = 3x^ e. f (x) = โ 3 x^5 + ex^ โ ฯ f. f (x) = x^1 /^2 (x^3 + 1/x)
