d/(dt)((x)(arctan(3/x)+arctan(9/x)))

Lösung

\frac{d}{d t} ((t) (arctan (\frac{3}{t}) + arctan (\frac{9}{t})))

arctan (\frac{3}{t}) + arctan (\frac{9}{t}) + \frac{t ( - 12 t ^{2} - 324 )}{( t ^{2} + 9 ) ( t ^{2} + 81 )}

Schritte zur Lösung

\frac{d}{d t} ((t) (arctan (\frac{3}{t}) + arctan (\frac{9}{t})))

Wende die Produktregel an:

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

f = t, g = arctan (\frac{3}{t}) + arctan (\frac{9}{t})

= \frac{d t}{d t} (arctan (\frac{3}{t}) + arctan (\frac{9}{t})) + \frac{d}{d t} (arctan (\frac{3}{t}) + arctan (\frac{9}{t})) t

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

\frac{d}{d t} (arctan (\frac{3}{t}) + arctan (\frac{9}{t})) = \frac{- 12 t ^{2} - 324}{( t ^{2} + 9 ) ( t ^{2} + 81 )}

= 1 \cdot (arctan (\frac{3}{t}) + arctan (\frac{9}{t})) + \frac{- 12 t ^{2} - 324}{( t ^{2} + 9 ) ( t ^{2} + 81 )} t

Vereinfache

1 \cdot (arctan (\frac{3}{t}) + arctan (\frac{9}{t})) + \frac{- 12 t ^{2} - 324}{( t ^{2} + 9 ) ( t ^{2} + 81 )} t : arctan (\frac{3}{t}) + arctan (\frac{9}{t}) + \frac{t ( - 12 t ^{2} - 324 )}{( t ^{2} + 9 ) ( t ^{2} + 81 )}

= arctan (\frac{3}{t}) + arctan (\frac{9}{t}) + \frac{t ( - 12 t ^{2} - 324 )}{( t ^{2} + 9 ) ( t ^{2} + 81 )}

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