AP Calculus AB/BC · Unit 2 · Topic 2.9
The Quotient Rule
Differentiate quotients while preserving denominator order and square.
1. Topic Focus
Define derivatives as limits, estimate slopes from representations, and establish the fundamental derivative rules.
This topic: Differentiate quotients while preserving denominator order and square.
2. Key Relationship
Read every symbol with its domain, direction, units, and hypotheses before applying the relationship.
3. Visual Connection
4. Worked Example
Differentiate (x²+1)/(x−1) and simplify only after applying the rule.
Write the governing relationship first, carry out the algebra cleanly, and finish with a sentence that answers the mathematical question.
5. Concept Development
If \(Q(x)=f(x)/g(x)\), where \(g(x)\ne0\), the quotient rule is
The numerator order matters: derivative of the numerator times the denominator, minus the numerator times derivative of the denominator. The original denominator is then squared.
| Position | Expression | Role |
|---|---|---|
| First numerator term | \(f'g\) | The top changes while the bottom keeps its current value |
| Second numerator term | \(-fg'\) | The bottom changes and is subtracted |
| Denominator | \(g^2\) | The entire original denominator is squared |
Why the rule has this form. Write \(Q=f/g\) as \(Qg=f\). Differentiating the product gives
Solving for \(Q'\) and substituting \(Q=f/g\),
This derivation explains both the subtraction and the squared denominator.
Why \(f'/g'\) is not valid. For \(x\ne0\), \(x^2/x=x\), whose derivative is 1. Dividing the derivatives would give \(2x/1=2x\), so differentiating the numerator and denominator separately cannot be the general rule.
Using values from a table. At \(x=a\), four values may be required:
Place the values by their labels before doing arithmetic. A table entry for \(g'(a)\) belongs in the second numerator term, not in the denominator.
Simplify first when it genuinely helps. A quotient of powers can often be rewritten with negative exponents, and polynomial factors may cancel. This can make the power rule shorter than the quotient rule. However, preserve the original domain: canceling a factor does not automatically define a previously missing point.
Horizontal tangents. A quotient has a horizontal tangent where its derivative equals zero. For a defined quotient-rule expression, this requires
while the original denominator must remain nonzero. A zero denominator is not a horizontal tangent; it is outside the function's domain.
Domain check. The derivative can only exist where the original quotient is defined and the needed component derivatives exist. Keep restrictions visible until the final answer, even if algebra cancels a factor in the formula.
6. Detailed Worked Example and Error Check
Example 1: Rational function. Let
With \(f=x^2+1\) and \(g=x-2\),
The unsimplified first line makes the quotient-rule order easy to audit.
Example 2: Trigonometric numerator. For \(x\ne0\),
Although \(\sin x/x\) has a finite limit at 0, the original function remains undefined there unless a separate extension is specified.
Example 3: Use a table of values. Suppose \(f(a)=6\), \(f'(a)=2\), \(g(a)=3\), and \(g'(a)=-1\). Then
The negative value of \(g'(a)\) makes the second contribution positive after subtraction.
Example 4: Quotient rule and tangent line. Let \(h(x)=(x+1)/(x-1)\). At \(x=2\), \(h(2)=3\), and
Since \(h'(2)=-2\), the tangent line is \(y-3=-2(x-2)\).
Example 5: Simplify while preserving a hole. For \(x\ne1\),
Therefore \(r'(x)=1\) for \(x\ne1\). The simplified formula does not create \(r(1)\), so the derivative is still not defined at the hole.
Example 6: A changing ratio in context. Density is \(D(t)=M(t)/V(t)\). At one instant, suppose \(M=10\) kg, \(V=4\) L, \(M'=0.6\) kg/min, and \(V'=0.1\) L/min. Then
Density is increasing at 0.0875 kilograms per liter per minute at that instant.
AP error check. Do not reverse the numerator order, omit the minus sign, square only part of the denominator, divide the two derivatives, cancel terms across addition, or accept a derivative value where the original denominator is zero.
7. AP Reasoning Routine
Identify the function structure, state the applicable rule, preserve notation and units, and check differentiability before interpreting a derivative.
- Identify the representation and requested quantity.
- State the rule or theorem and verify its conditions.
- Keep exact values until the final requested approximation.
- Interpret sign, units, interval, and context.
Apply the quotient rule and preserve all domain restrictions.
(a) Differentiate \(F(x)=(3x^2-1)/(x+2)\).
(b) If \(f(a)=4\), \(f'(a)=-2\), \(g(a)=5\), and \(g'(a)=3\), find \((f/g)'(a)\).
(c) Differentiate \(y=\sin x/e^x\) and simplify if helpful.
(d) Find the tangent line to \(h(x)=(x+1)/(x-1)\) at \(x=2\).
(e) Differentiate \((x^2-9)/(x-3)\), and explain what happens at \(x=3\).
(f) Find every horizontal tangent of \(q(x)=(x^2+1)/(x+1)\).
(g) A concentration is \(C(t)=S(t)/V(t)\). At an instant, \(S=12\) mg, \(V=3\) L, \(S'=0.9\) mg/min, and \(V'=0.2\) L/min. Find and interpret \(C'\).
(h) Use \(x^2/x=x\) for \(x\ne0\) to explain why \((f/g)'=f'/g'\) is false.
Check the solution
(a) \(F'(x)=[6x(x+2)-(3x^2-1)]/(x+2)^2\), for \(x\ne-2\).
(b) \((f/g)'(a)=[(-2)(5)-(4)(3)]/25=-22/25\).
(c) \(y'=[e^x\cos x-e^x\sin x]/e^{2x}=e^{-x}(\cos x-\sin x)\).
(d) \(h(2)=3\) and \(h'(2)=-2\), so \(y-3=-2(x-2)\).
(e) The quotient equals \(x+3\) for \(x\ne3\), so its derivative is 1 on that domain. The original function and derivative remain undefined at 3.
(f) The derivative numerator is \(2x(x+1)-(x^2+1)=x^2+2x-1\). Its zeros are \(x=-1\pm\sqrt2\), and both are in the domain.
(g) \(C'=[(0.9)(3)-(12)(0.2)]/9=1/30\) milligrams per liter per minute. Concentration is increasing at that rate.
(h) The simplified function has derivative 1, while dividing the derivatives gives \(2x\). Therefore the quotient of derivatives is not the derivative of a quotient.