In calculus, the derivative of any linear combination of functionss equals the same linear combination of the derivatives of the functions; this property is known as linearity of differentiation, the rule of linearity, or the superposition rule for differentiation. It is a fundamental property of the derivative that encapsulates in a single rule two simpler rules of differentiation, the sum rule (the derivative of the sum of two functions is the sum of the derivatives) and the constant factor rule (the derivative of a constant multiple of a function is the same constant multiple of the derivative). Thus it can be said that differentiation is linear, or the differential operator is a linear operator.

Statement and derivation

Let and be functions, with and constants. Now consider

By the sum rule in differentiation, this is

and by the constant factor rule in differentiation, this reduces to

Therefore,

Omitting the bracketss, this is often written as:

Detailed proofs/derivations from definition

We can prove the entire linearity principle at once, or, we can prove the individual steps (of constant factor and adding) individually. Here, both will be shown.

. (From a technical standpoint, the domain. One could also define both the second constant coefficient and the second function to be 0, where the domain of the second function is a superset of the first function, among other possibilities.)

. This would, when simplified, give us the difference rule for differentiation.

In the proofs/derivations below, above.

Linearity (directly)

.

.

By definition, we can see that

both individually exist. This allows us to use the coefficient law for limits to write

and

.

Sum

.

.

By definition, we can see that

exist. So, assuming that the derivatives exist, we can continue the above derivation

.

Difference

.

.

exist. So, assuming that the derivatives exist, we can continue the above derivation

.

Constant coefficient

.

.

By definition, we can see that:

Now, in order to use a limit law for constant coefficients to show that

exists.

exists, we can use the limit law and continue our proof.

.

See also

References