derivative of f(x^{-1})f(x)= 1/5 x+5,x=6

Lösung

\frac{d}{d x} (f (x)^{- 1}) f (x) = \frac{1}{5} x + 5, x = 6

f (x) = \frac{x ^{2} + 50 x + c _{1}}{10} x, f (x) = - \frac{x ^{2} + 50 x + c _{1}}{10} x

Schritte zur Lösung

\frac{d}{d x} ((f (x) x^{- 1}) f (x)) = \frac{1}{5} x + 5

Wenn

f^{^{'}} (x) = g (x)

dann

f (x) = \int g (x) d x

f (x) x^{- 1} f (x) = \int \frac{1}{5} x + 5 d x

\int \frac{1}{5} x + 5 d x = \frac{x ^{2}}{10} + 5 x + c_{1}

f (x) x^{- 1} f (x) = \frac{x ^{2}}{10} + 5 x + c_{1}

Isoliere

f (x) : f (x) = \frac{x ^{2} + 50 x + 10 c _{1}}{10} x, f (x) = - \frac{x ^{2} + 50 x + 10 c _{1}}{10} x

f (x) = \frac{x ^{2} + 50 x + 10 c _{1}}{10} x, f (x) = - \frac{x ^{2} + 50 x + 10 c _{1}}{10} x

Vereinfache

f (x) = \frac{x ^{2} + 50 x + c _{1}}{10} x, f (x) = - \frac{x ^{2} + 50 x + c _{1}}{10} x

x \frac{d y}{d x} = x^{2} y^{2}

\frac{d y}{d x} = y^{2} \cdot x^{3}

y^{\frac{5}{2}} \frac{d y}{d x} - y^{2} = - y^{\frac{7}{2}}

P^{^{'}} = k P (t)

x y^{^{'}} y + 4 = 0