Derivative of product — rule?
$(uv)' = u'v + uv'$
Primitive of $ rac{u'}{u}$ — function?
$oxed{ ext{Primitive} = ext{ln}|u| + C}$
Limit of $ ext{ln } x$ at zero?
$- ext{Infinity}$
Limit of $ ext{ln } x$ at infinity?
$+ ext{Infinity}$
Limit of $f(x)$ at zero or infinity?
Depends on the function's behavior near the point.
Limit of $ ext{ln } U(x)$ — when $U(x) o ext{?}$
$+ ext{Infinity}$ if $U(x) o+ ext{Infinity}$; $- ext{Infinity}$ if $U(x) o 0^+$; $ o ext{ln }k$ if $U(x) o k>0$.
Limit calculation techniques?
Use L'Hôpital, dominant terms, substitution, or algebraic simplification.
Limit of $x^eta ext{ln } x$ as $x o 0^+$?
$0$ for any real $eta$.
Limit of $ rac{ ext{ln } x}{x^eta}$ as $x o ext{?}$
$0$ as $x o + ext{Infinity}$ for $eta>0$.
Behavior of $ ext{ln } x$ near zero?
Tends to $- ext{Infinity}$ as $x o 0^+$.
Behavior of $ ext{ln } x$ at infinity?
Tends to $+ ext{Infinity}$ as $x o + ext{Infinity}$.
Limit of $ ext{ln } U(x)$ when $U(x) o 0^+$?
$- ext{Infinity}$.
Limit of $ ext{ln } U(x)$ when $U(x) o + ext{Infinity}$?
$+ ext{Infinity}$.
Limit of $ ext{ln } U(x)$ when $U(x) o k>0$?
$ ext{ln }k$.
Key property of $ ext{ln } x$?
Diverges to $- ext{Infinity}$ at zero and $+ ext{Infinity}$ at infinity.
Why use the primitive of $ rac{u'}{u}$?
To integrate functions involving ratios of derivatives and functions.
Teste tes connaissances avec un QCM de 8 questions sur LOGARITHMIC LIMITS AND DERIVATIVES.
1. What is the derivative of the product of two differentiable functions called?
2. What is the primitive (antiderivative) of the function $ rac{u'}{u}$, where $u$ is differentiable and non-zero?
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