Why Sums and Differences Have Simple Derivatives
A function may be built by adding or subtracting simpler functions. If we know how each part changes near an input, it is natural to ask whether the whole expression changes according to the sum or difference of those rates.
The answer is yes, provided the component functions are differentiable at the input in question. The reason is visible directly in the difference quotient: the change in a sum is the sum of the changes, and the limit of a finite sum is the sum of its limits.
Definitions and the Rules
For a function \(f\) defined on an interval around \(a\), its derivative at \(a\), when it exists, is the limit of its difference quotient:
In this definition, \(h\) is nonzero while the quotient is formed, and the limit is taken as \(h\) approaches zero. The derivative records the limiting rate of change at \(a\).
These rules apply at any point where both component derivatives exist. They do not require either function to be differentiable everywhere. To obtain a derivative function on an interval, the component functions must be differentiable at every point of that interval.
Why the Rules Work
The proof uses the derivative definition and the limit law that the limit of a sum of two functions is the sum of their limits, when both limits exist as finite numbers. Differentiability of \(f\) and \(g\) at \(a\) supplies exactly those finite limits.
Use the definition of the derivative for \(f+g\).
Expand the numerator and split the quotient into the difference quotients for \(f\) and \(g\).
Since both component limits exist, the sum limit is the sum of those limits.
In full, for \(h\ne 0\),
Because \(f\) and \(g\) are differentiable at \(a\), the two difference quotients on the right have finite limits \(f'(a)\) and \(g'(a)\). Therefore, taking the limit as \(h\to 0\) gives
This also proves that the derivative of the sum exists at \(a\). For the difference, the same calculation changes only the sign of the second component:
Both limits on the right exist, so the limit of their difference exists and equals \(f'(a)-g'(a)\). This proves the Difference Rule, including its existence condition.
Using the Rules with Several Terms
The rules can be applied repeatedly to a finite sum or difference. For example, if \(f_1,\ldots,f_n\) are all differentiable at \(a\), then their sum is differentiable there and
Likewise, parentheses can be handled by treating each grouped expression as a function, then applying the rules again. A negative sign in front of a function means its derivative is subtracted. The Constant Multiple Rule, established earlier, can also be used for numerical coefficients.
Worked Example: Differentiate a Sum of Two Functions
Let \(F(x)=x^2+3x\). The component functions are differentiable for every real \(x\). From the earlier results on powers, the identity function, and constant multiples, the derivatives of \(x^2\) and \(3x\) are \(2x\) and \(3\), respectively. Thus the Sum Rule gives
To check this result directly from the difference quotient, calculate the change in \(F\):
Taking the limit gives
This agrees with the result from the Sum Rule. At \(x=2\), for instance, \(F'(2)=2(2)+3=7\).
Worked Example: Differentiate a Difference
Consider \(G(x)=\dfrac{1}{x}-x^2+4\), defined for \(x\ne 0\). The reciprocal function is differentiable for \(x\ne0\), and \(x^2\) and the constant function are differentiable for every real \(x\). Apply the Difference Rule and the Sum Rule, using the earlier derivative results for these component functions:
For a direct check, write the quotient and simplify it for \(h\ne0\), with \(x+h\ne0\):
For fixed \(x\ne0\), \(x+h\ne0\) whenever \(h\) is sufficiently close to zero. Therefore,
The direct calculation confirms the derivative and also shows why the domain restriction \(x\ne0\) matters.
Worked Example: Combine Derivative Values at One Input
Suppose \(u\), \(v\), and \(w\) are differentiable at \(a=2\), and the available information is
Define \(H(x)=2u(x)-v(x)+w(x)\). By the Constant Multiple Rule and repeated use of the Sum and Difference Rules,
Check the calculation from the difference quotient. For \(h\ne0\), its numerator separates as follows:
As \(h\to0\), the three component quotients tend to \(4\), \(-3\), and \(6\). Hence the limit is \(2(4)-(-3)+6=17\), agreeing with the rule-based calculation. The function values at \(2\) are not needed to compute this derivative; the derivative values are what determine the limiting rate of change.
A Common Mistake: Multiplying Derivatives
A frequent error is to treat addition as if it required multiplying the component derivatives. For example, someone might write
This is not the Sum Rule. A concrete check exposes the error. Let \(f(x)=x^2\) and \(g(x)=x\). At \(x=2\), the actual derivative of their sum is
The incorrect multiplication would give \(f'(2)g'(2)=4(1)=4\), which is not \(5\). The correction is to match the operation on the functions: addition of functions uses addition of derivatives, and subtraction uses subtraction of derivatives.
When to Use These Rules
Before applying the rules, check that every component is differentiable at the input under consideration. If even one required component derivative does not exist there, these rules alone do not establish that the combined function is differentiable there.
- Keep the signs attached to the terms they precede.
- Use parentheses when subtracting a component derivative that contains multiple terms.
- For a finite expression, apply the rules repeatedly until each component derivative is known.
- State domain restrictions inherited from components, such as excluding inputs where a denominator is zero.
- Do not infer differentiability just because a function is written as a sum; the component differentiability conditions matter.
The practical value of the rules is that they let a complicated expression be differentiated in manageable pieces, once the component derivatives are known. Their proof also gives a useful model for understanding differentiation rules: start with the difference quotient, use algebra to expose known limits, and then apply the appropriate limit laws.
Check Your Understanding
Answer each question using the derivative definition or the rules as appropriate.
- If \(f\) and \(g\) are differentiable at \(a\), state the derivative of \(f+g\) at \(a\).
- Why can the limit of the sum of two difference quotients be split into the sum of their limits in the proof?
- If \(p'(3)=2\), \(q'(3)=-5\), and \(R(x)=p(x)-2q(x)\), find \(R'(3)\).
- For \(J(x)=x^2-4x+1\), identify which operation rule is needed to combine the component derivatives.
- A learner claims \((f-g)'(a)=f'(a)+g'(a)\). What sign correction is needed?
