derivative of e^{3x}(x^2+y^2)

解

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

e^{3 x} \cdot 3 (x^{2} + y^{2}) + (2 x + 2 y \frac{d y}{d x}) e^{3 x}

解答ステップ

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

積の法則を適用:

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

f = e^{3 x}, g = x^{2} + y^{2}

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

\frac{d}{d x} (e^{3 x}) = e^{3 x} \cdot 3

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

= e^{3 x} \cdot 3 (x^{2} + y^{2}) + (2 x + 2 y \frac{d y}{d x}) e^{3 x}