(\partial)/(\partial x)(e^{p(x)}(p(x)+p(x)sqrt(p(x))))

Lösung

\frac{\partial}{\partial x} (e^{p (x)} (p (x) + p (x) p (x)))

e^{p x} p (p x + p^{\frac{3}{2}} x^{\frac{3}{2}}) + (p + \frac{3 p ^{\frac{3}{2}} x}{2}) e^{p x}

Schritte zur Lösung

\frac{\partial}{\partial x} (e^{p (x)} (p (x) + p (x) p (x)))

Vereinfache

e^{p (x)} (p (x) + p (x) p (x)) : e^{p x} (p x + p^{\frac{3}{2}} x^{\frac{3}{2}})

= \frac{\partial}{\partial x} (e^{p x} (p x + p^{\frac{3}{2}} x^{\frac{3}{2}}))

Wende die Produktregel an:

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

f = e^{p x}, g = p x + p^{\frac{3}{2}} x^{\frac{3}{2}}

= \frac{\partial}{\partial x} (e^{p x}) (p x + p^{\frac{3}{2}} x^{\frac{3}{2}}) + \frac{\partial}{\partial x} (p x + p^{\frac{3}{2}} x^{\frac{3}{2}}) e^{p x}

\frac{\partial}{\partial x} (e^{p x}) = e^{p x} p

\frac{\partial}{\partial x} (p x + p^{\frac{3}{2}} x^{\frac{3}{2}}) = p + \frac{3 p ^{\frac{3}{2}} x}{2}

= e^{p x} p (p x + p^{\frac{3}{2}} x^{\frac{3}{2}}) + (p + \frac{3 p ^{\frac{3}{2}} x}{2}) e^{p x}

\int e^{a x + b} d x

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