derivative f(u)= u/(u+a/u)

Lösung

ableitung von

f (u) = \frac{u}{u + \frac{a}{u}}

\frac{2 a u}{( u ^{2} + a ) ^{2}}

Schritte zur Lösung

\frac{d}{d u} (\frac{u}{u + \frac{a}{u}})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d u}{d u} ( u + \frac{a}{u} ) - \frac{d}{d u} ( u + \frac{a}{u} ) u}{( u + \frac{a}{u} ) ^{2}}

\frac{d u}{d u} = 1

\frac{d}{d u} (u + \frac{a}{u}) = 1 - \frac{a}{u ^{2}}

= \frac{1 \cdot ( u + \frac{a}{u} ) - ( 1 - \frac{a}{u ^{2}} ) u}{( u + \frac{a}{u} ) ^{2}}

Vereinfache

\frac{1 \cdot ( u + \frac{a}{u} ) - ( 1 - \frac{a}{u ^{2}} ) u}{( u + \frac{a}{u} ) ^{2}} : \frac{2 a u}{( u ^{2} + a ) ^{2}}

= \frac{2 a u}{( u ^{2} + a ) ^{2}}

y^{^{'}} = - x (y + 1), y (0) = 2

\frac{\partial}{\partial y} (2 x + y)

\int 2 x (4 x^{2} - 10)^{2} d x

d er i v a t i v e f (x) = - 3 x^{2} + 5 x + 2

\int \frac{4}{( x ^{2} - 4 )} d x