Algebra of derivatives
If and are differentiable at , then , , and (when throughout) are differentiable at , with and .
Lemma 3.3
Proof idea
For the product rule, insert and rearrange:
As , the first factor tends to ; the middle factor tends to because is differentiable at and therefore continuous at ; and the last factor tends to . This gives .
The addition rule follows by splitting the difference quotient of into the difference quotients of and , and the reciprocal rule is the quotient rule with constant numerator .
Related
Stated in
- Lemma 3.3§3.1 Introduction