Linearity of differentiation

In calculus, the derivative of any linear combination of functions equals the same linear combination of the derivatives of the functions;^[1] this property is known as linearity of differentiation, the rule of linearity,^[2] or the superposition rule for differentiation.^[3] 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).^[4][5] Thus it can be said that the act of differentiation is linear, or the differential operator is a linear operator.^[6]

Statement and derivation

Let f and g be functions, with α and β constants. Now consider:

$\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) + \beta \cdot g(x) )$

By the sum rule in differentiation, this is:

$\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) ) + \frac{\mbox{d}}{\mbox{d} x} (\beta \cdot g(x))$

By the constant factor rule in differentiation, this reduces to:

$\alpha \cdot f'(x) + \beta \cdot g'(x)$

This in turn leads to:

$\frac{\mbox{d}}{\mbox{d} x}(\alpha \cdot f(x) + \beta \cdot g(x)) = \alpha \cdot f'(x) + \beta \cdot g'(x)$

Omitting the brackets, this is often written as:

$(\alpha \cdot f + \beta \cdot g)' = \alpha \cdot f'+ \beta \cdot g'$

References

1. Blank, Brian E.; Krantz, Steven George (2006), Calculus: Single Variable, Volume 1, Springer, p. 177, ISBN 9781931914598.
2. Strang, Gilbert (1991), Calculus, Volume 1, SIAM, pp. 71–72, ISBN 9780961408824.
3. Stroyan, K. D. (2014), Calculus Using Mathematica, Academic Press, p. 89, ISBN 9781483267975.
4. Estep, Donald (2002), "20.1 Linear Combinations of Functions", Practical Analysis in One Variable, Undergraduate Texts in Mathematics, Springer, pp. 259–260, ISBN 9780387954844.
5. Zorn, Paul (2010), Understanding Real Analysis, CRC Press, p. 184, ISBN 9781439894323.
6. Gockenbach, Mark S. (2011), Finite-Dimensional Linear Algebra, Discrete Mathematics and Its Applications, CRC Press, p. 103, ISBN 9781439815649.