What is the Sum Rule for derivatives?

Question

Evelyn Agard · Accepted Answer

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#

Evelyn Agard · Answer

undefined The Sum Rule for derivatives states that the derivative of the sum of two functions is equal to the sum of their derivatives. In mathematical notation, if $ f(x) $ and $ g(x) $ are two functions, then the derivative of their sum, denoted $ (f(x) + g(x))' $ or $ \frac{d}{dx}(f(x) + g(x)) $, is equal to the sum of their derivatives individually, i.e., $ f'(x) + g'(x) $. Mathematically, it can be expressed as:

$$ \frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x) $$

Emilia Aguilar · Answer

undefined The Sum Rule for derivatives states that if $ f(x) $ and $ g(x) $ are differentiable functions, then the derivative of their sum is equal to the sum of their derivatives:

$$ \frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x) $$

In other words, the derivative of the sum of two functions is equal to the sum of the derivatives of the individual functions. This rule holds true for any real-valued functions $ f(x) $ and $ g(x) $.