en.wikipedia.org/wiki/Chain_rule
2 corrections found
( f ( g ( a ) ) ) ′ \= q ( g ( a ) ) r ( a ) \= f ′ ( g ( a ) ) g ′ ( a ) .
This formula drops the composition symbol. The chain rule gives the derivative of the composite at a, `(f∘g)'(a)`, not a derivative of the single value `f(g(a))`.
Full reasoning
f(g(a)) is a single real number, not a function of the variable, so writing (f(g(a)))′ is not the chain-rule conclusion here. The standard chain rule statement is that if h(x)=f(g(x)), then the derivative at a is
h'(a) = (f∘g)'(a) = f'(g(a))\,g'(a).
That is exactly the form given in standard calculus texts. So the expression in the post is a notation error: the left-hand side should be (f∘g)'(a), not (f(g(a)))′.
2 sources
- 3.6 The Chain Rule - Calculus Volume 1 | OpenStax
For all x ... the derivative of the composite function h(x)=(f∘g)(x)=f(g(x)) is given by h′(x)=f′(g(x))g′(x).
- The Chain Rule
If f is differentiable at a and g is differentiable at [a], then the composite function is differentiable at a, and its derivative is ... In functional form, this is ...
written over a hundred years after Leibniz's discovery
Euler's main analysis books were written in 1745, 1748, and 1763, which are 69, 72, and 87 years after 1676—not over 100 years later.
Full reasoning
This date claim is off by more than a decade. Earlier in the post, Leibniz's discovery is dated to 1676. But the Euler Archive dates Euler's major analysis books as follows:
- Introductio in analysin infinitorum (vol. 1): Written Date 1745.
- Institutiones calculi differentialis: Written Date 1748.
- Institutionum calculi integralis volumen primum: Written Date 1763.
Relative to 1676, those are 69, 72, and 87 years later, respectively. Even using the publication dates (1748, 1755, 1768), the gaps are still only 72, 79, and 92 years. So "over a hundred years" is not correct for Euler's analysis books.
3 sources
- "Introductio in analysin infinitorum, volume 1" by Leonhard Euler
Published Date 1748 Written Date 1745.
- "Institutiones calculi differentialis cum eius usu in analysi finitorum" by Leonhard Euler
Published Date 1755 Written Date 1748.
- "Institutionum calculi integralis volumen primum" by Leonhard Euler
Published Date 1768 Written Date 1763.