Download Derivatives of Constant and Linear Functions

These notes closely follow the presentation of the material given in James Stewart’s
textbook Calculus, Concepts and Contexts (2nd edition). These notes are intended
primarily for in-class presentation and should not be regarded as a substitute for
thoroughly reading the textbook itself and working through the exercises therein.
Derivatives of Constant and Linear Functions
If c is a constant, then the derivative of the function fx  c (with domain all real
numbers) is f  x  0. In other words,
d c  0.
dx
Example Let f be the function fx  13 (with domain all real numbers). Find the
derivative of this function.
If m and b are constants, then the derivative of the function fx  mx  b (with
domain all real numbers) is f  x  m. In other words,
d mx  b  m.
dx
Example Let f be the function fx  5x  4 (with domain all real numbers). Find the
derivative of this function.
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Derivatives of Monic Monomial Functions
If n is a fixed positive integer, then the function fx  x n (with domain all real
numbers) is called a monic monomial function. (It is called a “monomial” because it
has only one term and it is called “monic” because the coefficient of this term is 1.)
The derivative of this function f  x  n  x n1 . In other words,
d x n   n  x n1 .
dx
This fact is sometimes referred to as the “Power Rule” of differentiation.
Example Find the derivative of the monic monomial function y  x 6 (with domain all
real numbers).
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The Constant Multiple and Sum Rules of
Differentiation
The Constant Multiple Rule of Differentiation
If f is a differentiable function and c is a constant, then
cf   cf  .
Example Find the derivative of the function y  4x 6 (with domain all real numbers).
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The Sum Rule of Differentiation
If f and g are differentiable functions, then
f  g   f   g  .
Example Find the derivative of the function fx  x 2  15.
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Derivatives of Polynomial Functions
By combining the Power, Constant Multiple, and Sum Rules of Differentiation, we
can find the derivative of any polynomial function.
Example For the derivative of the polynomial function fx  3x 2  6x  24 (with
domain all real numbers). A graph of f is shown below. Sketch the graph of f 
below the graph of f and convince yourself that these graphs make sense in relation
to each other.
-5
-4
-3
-2
-1
1
2
3
4
5
Graph of fx  3x 2  6x  24
-5
-4
-3
-2
-1
1
2
3
4
5
Graph of f  x  ________________
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Example Find the derivative of the polynomial function
y  6x 10  4x 7  2x 6  x 3  6x  865.
The Generalized Power Rule of Differentiation
Although we cannot prove it right now, the Power Rule
d x n   n  x n1
dx
is true even if the number n is not a positive integer. In fact, n can be any real number.
Example Find the derivatives of the functions fx  x , fx  1/x, and fx  4/x 3 .
(Hint: Each of these functions is really a “power” function in disguise.)
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Derivatives of Exponential Functions
If a is positive constant, then the function with domain all real numbers defined by
fx  a x is called the base a exponential function. For example, the base 2
exponential function is fx  2 x . The most important and useful exponential function
(at least for Calculus purposes) is the base e exponential function, fx  e x where e is
a certain irrational number that is approximately equal to 2. 71828. We will see in a
moment why base e is the preferable choice of a base when using exponential
functions to solve Calculus problems. (The reason, it turns out, is that its derivative is
“nicer” than the derivative of any other exponential function.)
The first thing we will do will be to observe that the derivative of an exponential
function, fx  a x , at any real number x is actually a constant multiple of a x .
Furthermore, the constant that multiplies a x is f  0. In other words, if f is the function
fx  a x and x is any real number, then
f  x  f  0  a x .
Here is the reason: For the function fx  a x and for a fixed real number x, we
have
fx  h  fx
f  x  lim
h0
h
xh
x
 lim a  a
h0
h
x
h
x
 lim a  a  a
h0
h
h
 lim a  1  a x .
h0
h
x
Since a is a constant, we may “pull it outside” of the above limit to obtain
h
f  x  lim a  1  a x .
h0
h
Finally, we observe that
h
0h
0
f0  h  f0
 f  0.
lim a  1  lim a  a  lim
h0
h0
h0
h
h
h
Thus
f  x  f  0  a x .
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Example For the exponential function fx  2 x , use numerical approximation (on your
calculator) to estimate the value of f  0. Then write an approximate formula for
the derivative of the function fx  2 x .
Answer: After doing some computations, we estimate that f  0  __________.
An approximate formula for the derivative of f is
d 2 x   _____________.
dx
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Example For the exponential function fx  3 x , use numerical approximation (on your
calculator) to estimate the value of f  0. Then write an approximate formula for
the derivative of the function fx  3 x .
Answer: After doing some computations, we estimate that f  0  __________.
An approximate formula for the derivative of f is
d 3 x   _____________.
dx
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For any exponential function, fx  a x , we have
f  x  f  0  a x .
The number f  0 depends on the base (a) that is being used. Furthermore, the
number f  0 increases (in continuous fashion) as the base a increases. We have seen
that if a  2, then f  0  0. 69, and that if a  3, then f  0  1. 098 6. This means that
there is a base, which we call e, that lies between 2 and 3 such that f  0 is exactly
equal to 1. The number e is defined in this way. In other words, the number e is the
number – the only number – such that
h
lim e  1  1.
h0
h
We can now see what makes base e so “nice” for Calculus purposes:
d e x   1  e x  e x .
dx
In other words, the derivative of the function fx  e x is simply the function itself!
Can you think of any other functions that are equal to their derivatives?
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