Use the transform .
I've been stumped trying to find the derivative .
By the chain, rule: .
Substituting and results in:
However, that cannot be the correct derivative. By inspecting their graphs, appears to have a critical (turning) point, yet there is no solution to .
Is there any reason for this apparent contradiction? Did I make a mistake?
Thanks in advance for any insight on this issue.
Just by the way - - Wolfram|Alpha does it with the chain rule for two inner functions...
... which doesn't mean we have to, of course, but I fancy this picture makes it palatable...
... is the double version of...
... the ordinary chain rule. As with that, straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed lines similarly but with respect to the (corresponding) dashed balloon expression which is (one of) the inner function(s) of the composite expression.
Shading shows that an expression has been treated as a constant on the way down (during differentiation).
You do, admittedly, need logarithmic differentiation to see the logic of the double-dashed differentiation, i.e. the logic of...
Don't integrate - balloontegrate!
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