What is the first d/(dt)(tln(t)) ?

The first d/(dt)(tln(t)) is ln(t)+1

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d/(dt)(tln(t))

\frac{d}{d t} (t ln (t))

ln (t) + 1

Solution steps

\frac{d}{d t} (t ln (t))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = t, g = ln (t)

= \frac{d t}{d t} ln (t) + \frac{d}{d t} (ln (t)) t

\frac{d t}{d t} = 1

\frac{d}{d t} (ln (t)) = \frac{1}{t}

= 1 \cdot ln (t) + \frac{1}{t} t

Simplify

1 \cdot ln (t) + \frac{1}{t} t : ln (t) + 1

= ln (t) + 1

x \to 0 + lim (\frac{x}{ln ( x )})

f (x) = x^{2} + 4 x - 5, - 3, 3

x \to 3 lim (\frac{5 x}{x - 3})

\frac{d}{d x} (ln (\frac{3 x}{2 x + 1}))

\int \frac{1}{( x + 2 ) ( x + 7 ) ( 2 x - 5 ) ( x + 1 ) ^{2}} d x