2.1. ELEMENTARY DERIVATIVE RULES
91
Constant Multiples and Sums of Functions
Of course, most of the functions we encounter in mathematics are more complicated than
being simply constant, a power of a variable, or a base raised to a variable power. In this
section and several following, we will learn how to quickly compute the derivative of a
function constructed as an algebraic combination of basic functions. For instance, we’ d
like to be able to understand how to take the derivative of a polynomial function such as
p(t) = 3t 5 − 7t 4 + t 2 − 9, which is a function made up of constant multiples and sums of
powers of t. To that end, we develop two new rules: the Constant Multiple Rule and the
Sum Rule.
Say we have a function y = f (x) whose derivative formula is known. How is the
derivative of y = k f (x) related to the derivative of the original function? Recall that when
we multiply a function by a constant k, we vertically stretch the graph by a factor of |k|
(and reflect the graph across y = 0 if k < 0). This vertical stretch affects the slope of the
graph, making the slope of the function y = k f (x) be k times as steep as the slope of
y = f (x). In terms of the derivative, this is essentially saying that when we multiply a
function by a factor of k, we change the value of its derivative by a factor of k as well.
Thus 2 , the Constant Multiple Rule holds:
The Constant Multiple Rule: For any real number k, if f (x) is a differentiable
function with derivative f ′ (x), then
d
dx [k f (x)] = k f ′ (x).
In words, this rule says that “the derivative of a constant times a function is the
constant times the derivative of the function.” For example, if g(t) = 3 · 5 t , we have
g ′ (t) = 3 · 5 t ln(5). Similarly,
d
dz [5z −2 ] = 5(−2z −3 ).
Next we examine what happens when we take a sum of two functions. If we have
y = f (x) and y = g(x), we can compute a new function y = ( f + g)(x) by adding the
outputs of the two functions: ( f + g)(x) = f (x) + g(x). Not only does this result in the
value of the new function being the sum of the values of the two known functions, but also
the slope of the new function is the sum of the slopes of the known functions. Therefore 3 ,
we arrive at the following Sum Rule for derivatives:
The Sum Rule: If f (x) and g(x) are differentiable functions with derivatives f ′ (x)
and g ′ (x) respectively, then
d
dx [ f (x) + g(x)] = f ′ (x) + g ′ (x).
In words, the Sum Rule tells us that “the derivative of a sum is the sum of the
derivatives.” It also tells us that any time we take a sum of two differentiable functions, the
result must also be differentiable. Furthermore, because we can view the difference function
y = ( f − g)(x) = f (x) − g(x) as y = f (x) + (−1 · g(x)), the Sum Rule and Constant Multiple
2 The Constant Multiple Rule can be formally proved as a consequence of properties of limits, using the
limit definition of the derivative.
3 Like the Constant Multiple Rule, the Sum Rule can be formally proved as a consequence of properties of
limits, using the limit definition of the derivative.
91
Constant Multiples and Sums of Functions
Of course, most of the functions we encounter in mathematics are more complicated than
being simply constant, a power of a variable, or a base raised to a variable power. In this
section and several following, we will learn how to quickly compute the derivative of a
function constructed as an algebraic combination of basic functions. For instance, we’ d
like to be able to understand how to take the derivative of a polynomial function such as
p(t) = 3t 5 − 7t 4 + t 2 − 9, which is a function made up of constant multiples and sums of
powers of t. To that end, we develop two new rules: the Constant Multiple Rule and the
Sum Rule.
Say we have a function y = f (x) whose derivative formula is known. How is the
derivative of y = k f (x) related to the derivative of the original function? Recall that when
we multiply a function by a constant k, we vertically stretch the graph by a factor of |k|
(and reflect the graph across y = 0 if k < 0). This vertical stretch affects the slope of the
graph, making the slope of the function y = k f (x) be k times as steep as the slope of
y = f (x). In terms of the derivative, this is essentially saying that when we multiply a
function by a factor of k, we change the value of its derivative by a factor of k as well.
Thus 2 , the Constant Multiple Rule holds:
The Constant Multiple Rule: For any real number k, if f (x) is a differentiable
function with derivative f ′ (x), then
d
dx [k f (x)] = k f ′ (x).
In words, this rule says that “the derivative of a constant times a function is the
constant times the derivative of the function.” For example, if g(t) = 3 · 5 t , we have
g ′ (t) = 3 · 5 t ln(5). Similarly,
d
dz [5z −2 ] = 5(−2z −3 ).
Next we examine what happens when we take a sum of two functions. If we have
y = f (x) and y = g(x), we can compute a new function y = ( f + g)(x) by adding the
outputs of the two functions: ( f + g)(x) = f (x) + g(x). Not only does this result in the
value of the new function being the sum of the values of the two known functions, but also
the slope of the new function is the sum of the slopes of the known functions. Therefore 3 ,
we arrive at the following Sum Rule for derivatives:
The Sum Rule: If f (x) and g(x) are differentiable functions with derivatives f ′ (x)
and g ′ (x) respectively, then
d
dx [ f (x) + g(x)] = f ′ (x) + g ′ (x).
In words, the Sum Rule tells us that “the derivative of a sum is the sum of the
derivatives.” It also tells us that any time we take a sum of two differentiable functions, the
result must also be differentiable. Furthermore, because we can view the difference function
y = ( f − g)(x) = f (x) − g(x) as y = f (x) + (−1 · g(x)), the Sum Rule and Constant Multiple
2 The Constant Multiple Rule can be formally proved as a consequence of properties of limits, using the
limit definition of the derivative.
3 Like the Constant Multiple Rule, the Sum Rule can be formally proved as a consequence of properties of
limits, using the limit definition of the derivative.