Question

What is the Sum Rule for derivatives?

Answer 1

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(x) = g(x) + h(x)`

we can express the derivative of `f(x)`, `f'(x)`, as

`f'(x) = g'(x) + h'(x)`.

For an example, consider a cubic function:

`f(x) = Ax^3 + Bx^2 + Cx + 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.

`d/dx(c*f(x)) = c*((df)/dx)` and `d/dx(c) = 0`, where `c` represents any constant.

The third is the Power Rule, which states that for a quantity `x^n`, `d/dx(x^n) = nx^(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 `d/dx x = 1`

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

`d/dx f(x) = d/dx[Ax^3 + Bx^2 + Cx +D]`

`= d/dx Ax^3 + d/dx Bx^2 + d/dx Cx + d/dx D`

`= A(d/dx x^3) + B(d/dx x^2) + C(d/dx x) + D(d/dx 1)`

`= A(3x^2) + B(2x) + C(1) + 0`

`(df)/dx = 3Ax^2 + 2Bx +C`