Deriving the Constant-Multiple and Sum Rules

Why These Rules Matter

The difference quotient defines a derivative, but using it from scratch for every expression can involve repeated algebra. The constant-multiple and sum rules explain how differentiation interacts with two basic operations: multiplying a function by a constant, and adding functions.

These rules are not just shortcuts. They follow from the definition of the derivative, provided the functions involved are differentiable at the point in question. That condition matters: a rule for combining derivatives cannot create a derivative where one of the component functions has none.

The Difference Quotient and the Conditions

For a function \(f\), the derivative at an interior point \(a\) of its domain is defined by the limit of its difference quotient, when that limit exists:

The increment \(h\) is nonzero while the quotient is formed, and it approaches zero in the limit. If this limit exists as a finite real number, \(f\) is differentiable at \(a\). The rules below apply at any point where the functions in them are differentiable. In particular, to apply a rule at \(a\), the component functions must be defined near \(a\) and have derivatives there.

Deriving the Constant-Multiple Rule

Let \(c\) be a constant and suppose \(f\) is differentiable at \(a\). Apply the difference quotient definition to the function \(cf\). Since \(c\) does not depend on \(h\), it can be factored out of the quotient:

The third equality uses the constant-multiple law for limits: if a limit exists, multiplying every value in the expression by a fixed constant multiplies the limit by that constant. The last limit exists because \(f\) is differentiable at \(a\). Thus the rule follows directly from the definition, rather than from a pattern observed in examples.

The constant may be positive, negative, or zero. If \(c=0\), the function \(cf\) is the zero function, whose derivative is zero; the formula gives \(0f'(a)=0\), as expected.

Deriving the Sum Rule

Suppose \(f\) and \(g\) are both differentiable at \(a\). In the difference quotient for \(f+g\), separate the change in the sum into the change in each function:

The third equality uses the limit law for sums. It is valid here because differentiability of \(f\) and \(g\) guarantees that both difference-quotient limits exist. If either limit does not exist, this derivation cannot be used to assert that the sum has a derivative.

The difference rule is a consequence of these two results: subtracting \(g\) is the same as adding \((-1)g\). Therefore, if \(f\) and \(g\) are differentiable at \(a\),

A Basic Derivative to Use with the Rules

The identity function \(I(x)=x\) has derivative \(1\), since its difference quotient simplifies to \(1\) for every nonzero \(h\):

A constant function has derivative zero, as established in the earlier material on the derivative of a constant function. These two facts let us apply the new rules to linear functions, without needing any additional derivative rules.

Worked Examples

Why the Differentiability Conditions Matter

The algebra in the derivations splits a difference quotient into component quotients. To take the limit of their sum, the component limits must exist. This is why the sum rule requires both component functions to be differentiable at the point. The same condition applies to the difference rule and to the constant-multiple rule's function.

These rules give a reliable way to combine derivatives that already exist. They do not say that every sum or multiple is differentiable everywhere regardless of its components. At a point where a component derivative fails to exist, the rule alone gives no conclusion about the combined function; further analysis would be needed.

Common Mistake: Keeping a Constant in the Derivative

A frequent error is to treat a constant term as though differentiation leaves it unchanged. For example, an incorrect calculation might say

This confuses the original constant with its derivative. The constant function \(-7\) has derivative \(0\), because its output does not change when \(x\) changes. The correct calculation is

The difference quotient confirms the correction: the constant terms cancel in \(F(x+h)-F(x)\), leaving only \(4h\), and dividing by \(h\) gives \(4\).

What These Rules Establish

The constant-multiple and sum rules allow derivatives to be combined linearly: multiplying a differentiable function by a constant multiplies its derivative by that constant, and adding differentiable functions adds their derivatives. Subtraction follows by multiplying one function by \(-1\). In the examples here, the rules apply to linear functions because their derivatives follow directly from the identity and constant functions.

The scope is deliberate. These rules alone do not yet determine derivatives of higher powers such as \(x^2\) or \(x^3\). To differentiate such terms, another result is needed. For now, the key habits are to check that the component functions are differentiable at the point, apply the rules with constants handled correctly, and use a difference quotient when a direct check is useful.