Proving the Sum & Difference Rules for Derivatives

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Proving a derivative rule leads to a better understanding of differentiating. In this lesson we show how to prove the sum and difference rules for derivatives.

Derivatives, Rules and Proofs

Language can be confusing and some things should not be taken literally. Do you see Derifun staring at tonight's desert? The homework assignment is to develop some proofs for two derivative rules and someone told Derifun, the 'proof is in the pudding'.

We can help Derifun with the math but first we have to get ourselves clear of the refrigerator.

Where is the proof?

Fortunately, it's straightforward proving the sum and difference rules for derivatives as we will show in this lesson. Finding the proof in the pudding would be a lot messier!

Derivative of the Sum or Difference of Two Functions

Derifun asks for a quick review of derivative notation. In one line you write:

Derivative notation

In words: y prime is the same as f prime of x which is the same as d by dx f of x. Using a 'prime' for the derivative is sometimes referred to as prime notation. The last part with the 'lim' is the definition of the derivative.

Generally, when proving something in math, we use definitions and previously proven information. For example, in this lesson we use the definition of the derivative to prove the sum and differences results. But Derifun has more questions.

What if we had a function called h(x) and wanted to write the derivative? Answer: we would need h(x + Δx) - h(x) for the numerator.

What if h(x) is actually the sum of f(x) and g(x)? Answer: No problem! If h(x) is f(x) + g(x), then h(x + Δx) is f(x + Δx) + g(x + Δx).

We now have all the pieces to prove the derivative rule involving f(x) + g(x).


Focusing on the right-hand-side, we decide to group the 'f' terms and the 'g' terms:

Numerator regrouping

A property of limits is the limit of a sum is the sum of the limits. This is the part of the proof where we use previously proven information. We now write the limit of the expression on the right as the sum of two limits.

Sum of limits

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