derivative of x(49-x^2^{1/2})

Lösung

\frac{d}{d x} (x (49 - x^{2})^{\frac{1}{2}})

\frac{- 2 x ^{2} + 49}{( 49 - x ^{2} ) ^{\frac{1}{2}}}

Schritte zur Lösung

\frac{d}{d x} (x (49 - x^{2})^{\frac{1}{2}})

Wende die Produktregel an:

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

f = x, g = (49 - x^{2})^{\frac{1}{2}}

= \frac{d x}{d x} (49 - x^{2})^{\frac{1}{2}} + \frac{d}{d x} ((49 - x^{2})^{\frac{1}{2}}) x

\frac{d x}{d x} = 1

\frac{d}{d x} ((49 - x^{2})^{\frac{1}{2}}) = - \frac{x}{( 49 - x ^{2} ) ^{\frac{1}{2}}}

= 1 \cdot (49 - x^{2})^{\frac{1}{2}} + (- \frac{x}{( 49 - x ^{2} ) ^{\frac{1}{2}}}) x

Vereinfache

1 \cdot (49 - x^{2})^{\frac{1}{2}} + (- \frac{x}{( 49 - x ^{2} ) ^{\frac{1}{2}}}) x : \frac{- 2 x ^{2} + 49}{( 49 - x ^{2} ) ^{\frac{1}{2}}}

= \frac{- 2 x ^{2} + 49}{( 49 - x ^{2} ) ^{\frac{1}{2}}}

\int \frac{sin ( x ) cos ( x )}{4 + sin ^{4} ( x )} d x

\frac{d}{d x} (\frac{x + 2}{3})

x \to 0 lim (\frac{tan ( π x )}{x})

\frac{d}{d x} (e^{y z (x)})

x \to 2 lim ([4 f (- 8) + g (2)])