# What is the Sum Rule for derivatives?

Aug 29, 2014

The sum rule for derivatives states that the derivative of a sum is equal to the sum of the derivatives.

In symbols, this means that for

$f \left(x\right) = g \left(x\right) + h \left(x\right)$

we can express the derivative of $f \left(x\right)$, $f ' \left(x\right)$, as

$f ' \left(x\right) = g ' \left(x\right) + h ' \left(x\right)$.

For an example, consider a cubic function:

$f \left(x\right) = A {x}^{3} + B {x}^{2} + C x + D .$

Note that A, B, C, and D are all constants. Now we will make use of three other basic properties, two of which are illustrated together below, without proof.

$\frac{d}{\mathrm{dx}} \left(c \cdot f \left(x\right)\right) = c \cdot \left(\frac{\mathrm{df}}{\mathrm{dx}}\right)$ and $\frac{d}{\mathrm{dx}} \left(c\right) = 0$, where $c$ represents any constant.

The third is the Power Rule, which states that for a quantity ${x}^{n}$, $\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n {x}^{n - 1}$. This will also be accepted here without proof, in interests of brevity. Note that for the case $n = 1$, we would be taking the derivative of x with respect to x, which would inherently be one. Thus $\frac{d}{\mathrm{dx}} x = 1$

Using all four of these properties, we can find the derivative of our cubic expression.

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{d}{\mathrm{dx}} \left[A {x}^{3} + B {x}^{2} + C x + D\right]$

$= \frac{d}{\mathrm{dx}} A {x}^{3} + \frac{d}{\mathrm{dx}} B {x}^{2} + \frac{d}{\mathrm{dx}} C x + \frac{d}{\mathrm{dx}} D$

$= A \left(\frac{d}{\mathrm{dx}} {x}^{3}\right) + B \left(\frac{d}{\mathrm{dx}} {x}^{2}\right) + C \left(\frac{d}{\mathrm{dx}} x\right) + D \left(\frac{d}{\mathrm{dx}} 1\right)$

$= A \left(3 {x}^{2}\right) + B \left(2 x\right) + C \left(1\right) + 0$

$\frac{\mathrm{df}}{\mathrm{dx}} = 3 A {x}^{2} + 2 B x + C$