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2.3. THE PRODUCT AND QUOTIENT RULES
2.3 The product and quotient rules
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How does the algebraic structure of a function direct us in computing its derivative
using shortcut rules?
• How do we compute the derivative of a product of two basic functions in terms of
the derivatives of the basic functions?
• How do we compute the derivative of a quotient of two basic functions in terms of
the derivatives of the basic functions?
• How do the product and quotient rules combine with the sum and constant
multiple rules to expand the library of functions we can quickly differentiate?
Introduction
So far, the basic functions we know how to differentiate include power functions (x n ),
exponential functions (a x ), and the two fundamental trigonometric functions (sin(x) and
cos(x)). With the sum rule and constant multiple rules, we can also compute the derivative
of combined functions such as
f (x) = 7x
11 − 4 · 9
x + π sin(x) −
√
3 cos(x),
because the function f is fundamentally a sum of basic functions. Indeed, we can now
quickly say that f ′ (x) = 77x 10 − 4 · 9 x ln(9) + π cos(x) +
√
3 sin(x).
But we can of course combine basic functions in ways other than multiplying them by
constants and taking sums and differences. For example, we could consider the function
that results from a product of two basic functions, such as
p(z) = z
3 cos(z),
or another that is generated by the quotient of two basic functions, one like
q(t) =
sin(t)
2 t .
While the derivative of a sum is the sum of the derivatives, it turns out that the rules for
computing derivatives of products and quotients are more complicated. In what follows we
explore why this is the case, what the product and quotient rules actually say, and work to
expand our repertoire of functions we can easily differentiate. To start, Preview Activity 2.3
2.3. THE PRODUCT AND QUOTIENT RULES
2.3 The product and quotient rules
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How does the algebraic structure of a function direct us in computing its derivative
using shortcut rules?
• How do we compute the derivative of a product of two basic functions in terms of
the derivatives of the basic functions?
• How do we compute the derivative of a quotient of two basic functions in terms of
the derivatives of the basic functions?
• How do the product and quotient rules combine with the sum and constant
multiple rules to expand the library of functions we can quickly differentiate?
Introduction
So far, the basic functions we know how to differentiate include power functions (x n ),
exponential functions (a x ), and the two fundamental trigonometric functions (sin(x) and
cos(x)). With the sum rule and constant multiple rules, we can also compute the derivative
of combined functions such as
f (x) = 7x
11 − 4 · 9
x + π sin(x) −
√
3 cos(x),
because the function f is fundamentally a sum of basic functions. Indeed, we can now
quickly say that f ′ (x) = 77x 10 − 4 · 9 x ln(9) + π cos(x) +
√
3 sin(x).
But we can of course combine basic functions in ways other than multiplying them by
constants and taking sums and differences. For example, we could consider the function
that results from a product of two basic functions, such as
p(z) = z
3 cos(z),
or another that is generated by the quotient of two basic functions, one like
q(t) =
sin(t)
2 t .
While the derivative of a sum is the sum of the derivatives, it turns out that the rules for
computing derivatives of products and quotients are more complicated. In what follows we
explore why this is the case, what the product and quotient rules actually say, and work to
expand our repertoire of functions we can easily differentiate. To start, Preview Activity 2.3