How to Apply the Derivative Sum Rule

In calculus, there are a few fundamental basic rules that every student must know for differentiation. These differentiation rules are labeled in calculus parlance, including the constant function rule, the power rule, the multiple rule and the product rule quotient rule. The Sum Rule is the easiest to remember, and it's used more often than the other derivative rules.

Instructions

1. Remember that the sum rule says that the derivative of a sum of two numbers is the sum of their derivatives.

2. Note this rule: If two functions, "u" and "v," are differentiable for "x" then the sum of "u + v" is differentiable at every point where "u" and "v" are differentiable. In other words, (d/dx)(u + v) = du/dx + dv/dx.

3. Apply this rule for the equation: y = x^3 + 9x, dy/dx = (d/dx)(x^3) + (d/dx)(9x) = 3x^2 + 9.

4. Prove the rule: f(x) = u(x) + v(x). (d/dx)[u(x) + v(x)] equals the limit of h as it goes to "0" for [u(x + h) + v(x + h)] -[u(x) + v(x)] all over h. That equals the limit as h goes to '0' for [(u(x + h) - u(x))/h + (v(x + h) - v(x))/h], which equals the limit as h goes to '0' for [(u(x + h) - u(x))/h + the limit as h goes to '0' for (v(x + h) - v(x))/h] = du/dx + dv/dx.

5. Understand the derivative sum rule always work for all calculus derivative problems.