implicit (dy)/(dx),(xy)^{1/2}=1+x^2y

Lösung

implizite ableitung

\frac{d y}{d x}, (x y)^{\frac{1}{2}} = 1 + x^{2} y

\frac{d y}{d x} = \frac{4 x ^{\frac{3}{2}} y ^{\frac{3}{2}} - y}{x - 2 x ^{\frac{5}{2}} y ^{\frac{1}{2}}}

Schritte zur Lösung

(x y)^{\frac{1}{2}} = 1 + x^{2} y

Behandle

y

wie

y (x)

Beide Seiten differenzieren:

\frac{y + x \frac{d y}{d x}}{2 ( x y ) ^{\frac{1}{2}}} = x^{2} \frac{d y}{d x} + 2 x y

Isoliere

\frac{d y}{d x} : \frac{d y}{d x} = \frac{4 x ^{\frac{3}{2}} y ^{\frac{3}{2}} - y}{x - 2 x ^{\frac{5}{2}} y ^{\frac{1}{2}}}

\frac{d y}{d x} = \frac{4 x ^{\frac{3}{2}} y ^{\frac{3}{2}} - y}{x - 2 x ^{\frac{5}{2}} y ^{\frac{1}{2}}}

y^{^{''}} + 4 y^{^{'}} + 4 y = \frac{ln ( t )}{t} e^{- 2 t}

d er i v a t i v e (e^{x})^{2}

\int \frac{27}{w ^{2} 9 - w ^{2}} d w

\int \frac{( e ^{- x} + 6 ) ^{2}}{e ^{x}} d x

\frac{d}{d x} (5^{x} ln (x))