SME
Calculus I · Problem type
The product and quotient rules from a table
Use this for “f(2) = 3, f′(2) = −1, g(2) = 4, g′(2) = 5. Find (fg)′(2) and (f/g)′(2).”
Diagram
In Δt = 0.15 the area grows by u′Δt × v + u × v′Δt = −0.6 + 2.25, plus the corner u′v′Δt² = −0.1125, too small to count as Δt → 0.
(uv)′ = u′v + uv′
(uv)′ = (−1) × 4 + 3 × 5 = −4 + 15 = 11
(
u
v
)′
=
u′v − uv′
v
2
=
−4 − 15
16
(
u
v
)′
=
−1.188
This is an example. Type your own values to start a new problem.
u
=
u
,
u′
=
u′
,
v
=
v
,
v′
=
v′
(uv)′
=
u′
×
v
+
u
×
v′
=
P′
(u/v)′
=
×
−
×
(
)
2
=
Q′
Formulas
P′
=
u
′
×
v
+
u
×
v
′
11 = −1 × 4 + 3 × 5
Q′
=
u
′ ×
v
−
u
×
v
′
v
2
−1.1875
=
−1 × 4 − 3 × 5
4
2
Assumptions
•
u, u′, v and v′ are the values of two functions (f and g in most questions) and their slopes at one point x = a, read from a table.
•
For (g ÷ f)′, type g’s values as u and f’s as v. For x·f(x), u = x: type u = a and u′ = 1.
•
Product rule: (uv)′ = u′v + uv′. In a short time Δt the u-by-v rectangle grows by strips u′Δt by v and u by v′Δt. The corner u′v′Δt² shrinks faster, adding nothing.
•
Quotient rule: (u ÷ v)′ = (u′v − uv′) ÷ v², which needs v ≠ 0 at a.
Step-by-step
WE KNOW
u = 3, u′ = −1, v = 4, v′ = 5
FIND
slope of the product uv (P′), slope of the quotient u ÷ v (Q′)
Step 1 · Find slope of the product uv (P′)
Use P′ = u′ × v + u × v′
Product rule: each strip is one factor’s slope times the other factor.
P′ = −1 × 4 + 3 × 5
P′ = −4 + 15
P′ = 11
Step 2 · Find slope of the quotient u ÷ v (Q′)
Use Q′ = (u′ × v − u × v′) ÷ v²
Quotient rule: bottom times the top’s slope, minus top times the bottom’s slope, all over the bottom squared.
Q′
=
−1 × 4 − 3 × 5
4
2
Q′
=
−4 − 15
16
Q′
=
−
19
16
Q′ = −1.1875
Check
Put the numbers back in. Both sides should match.
11 = −1 × 4 + 3 × 5 ✓
−1.1875
=
−1 × 4 − 3 × 5
4
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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