derivative of arctan({g}(x)x)

Lösung

\frac{d}{d x} (arctan (g (x)) x)

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

Schritte zur Lösung

\frac{d}{d x} (arctan (g (x)) x)

Wende die Produktregel an:

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

f = arctan (g (x)), g = x

= \frac{d}{d x} (arctan (g (x))) x + \frac{d x}{d x} arctan (g (x))

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

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

= \frac{\frac{d}{d x} ( g ( x ) )}{g ( x ) ^{2} + 1} x + 1 \cdot arctan (g (x))

Vereinfache

\frac{\frac{d}{d x} ( g ( x ) )}{g ( x ) ^{2} + 1} x + 1 \cdot arctan (g (x)) : \frac{x \frac{d}{d x} ( g ( x ) )}{g ( x ) ^{2} + 1} + arctan (g (x))

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

t a y l or x, x = 1

t an g e n t y = 3 x^{3} + x^{2} - 8 x + 4, (- 2, 0)

\int \frac{3 x}{7 - x ^{2}} d x

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

\int sin (θ) + cos (θ) d θ