(\partial)/(\partial x)(5xln(x^2+y))

Lösung

\frac{\partial}{\partial x} (5 x ln (x^{2} + y))

5 (ln (x^{2} + y) + \frac{2 x ^{2}}{x ^{2} + y})

Schritte zur Lösung

\frac{\partial}{\partial x} (5 x ln (x^{2} + y))

Behandle

y

als Konstante

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= 5 \frac{\partial}{\partial x} (x ln (x^{2} + y))

Wende die Produktregel an:

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

f = x, g = ln (x^{2} + y)

= 5 (\frac{\partial}{\partial x} (x) ln (x^{2} + y) + \frac{\partial}{\partial x} (ln (x^{2} + y)) x)

\frac{\partial}{\partial x} (x) = 1

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

= 5 (1 \cdot ln (x^{2} + y) + \frac{2 x}{x ^{2} + y} x)

Vereinfache

5 (1 \cdot ln (x^{2} + y) + \frac{2 x}{x ^{2} + y} x) : 5 (ln (x^{2} + y) + \frac{2 x ^{2}}{x ^{2} + y})

= 5 (ln (x^{2} + y) + \frac{2 x ^{2}}{x ^{2} + y})

x \to 0 lim (x^{2} - 6)

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