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Proofs of the Sum, Difference, Product, and Quotient RulesThe Sum RuleWe begin with The Difference RuleIn the last lesson we showed that The Product RuleWorking out The Quotient RuleGiven
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. We can write 
. The limit can be rewritten as 
, or 
. The fraction can be split up into 
. We then apply the sum law to split the limit into 
, which is simply 
, so 
: the derivative of a sum is the sum of the derivatives.
(this is a degenerate case of the Product Rule, proven below), so we can find 
simply by regarding the second term as a constant of ‒1 times 
, so the difference rule is simply 
; the derivative of a difference is the difference of the derivatives.
requires a bit of ingenuity. We begin the differentiation by writing the problem as a difference quotient: 
. Now, we play with the numerator, making the limit 
. This is valid, since we are both adding and subtracting the term 
for a zero net change. We factor the numerator to give 
. Now we split the fraction and split the limit, just as we did to prove the sum rule; the expression left is 
. We know that 
since 
from the limit since it does not change as 
. Now we have 
. The two limits each represent derivatives, so this expression is equal to 
. Therefore the product rule states that 
. (The constant multiple rule is a degenerate case of the product rule, since the derivative of a constant is zero.)
, we know that 
, a limit which can of course be written as 
. We combine the two parts of the second term into one fraction, so we have 
. Now we use a similar technique to that used in solving for the product rule: we add and subtract 
to and from the numerator. Now the limit is 
, which we factor to give 
. Now we move the 
around a bit: 
, which we rewrite as 
. It’s apparent now that there are some difference quotients in there, so we can make the limit into 
. And since 
, we rewrite the denominator to give 
.