derivative of x^{1/2}(3x^2-35x+90)

Lösung

\frac{d}{d x} (x^{\frac{1}{2}} (3 x^{2} - 35 x + 90))

\frac{15 x ^{2} - 105 x + 90}{2 x ^{\frac{1}{2}}}

Schritte zur Lösung

\frac{d}{d x} (x^{\frac{1}{2}} (3 x^{2} - 35 x + 90))

Wende die Produktregel an:

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

f = x^{\frac{1}{2}}, g = 3 x^{2} - 35 x + 90

= \frac{d}{d x} (x^{\frac{1}{2}}) (3 x^{2} - 35 x + 90) + \frac{d}{d x} (3 x^{2} - 35 x + 90) x^{\frac{1}{2}}

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

\frac{d}{d x} (3 x^{2} - 35 x + 90) = 6 x - 35

= \frac{1}{2 x ^{\frac{1}{2}}} (3 x^{2} - 35 x + 90) + (6 x - 35) x^{\frac{1}{2}}

Vereinfache

\frac{1}{2 x ^{\frac{1}{2}}} (3 x^{2} - 35 x + 90) + (6 x - 35) x^{\frac{1}{2}} : \frac{15 x ^{2} - 105 x + 90}{2 x ^{\frac{1}{2}}}

= \frac{15 x ^{2} - 105 x + 90}{2 x ^{\frac{1}{2}}}

\frac{d ^{2}}{d x ^{2}} (3 (x^{2} - 1)^{2})

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

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

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\frac{d}{d θ} (sin^{2} (\frac{θ}{2}))