(\partial)/(\partial x)(pi*xsqrt(x^2+y^2))

解

\frac{\partial}{\partial x} (π \cdot x x^{2} + y^{2})

\frac{π ( 2 x ^{2} + y ^{2} )}{x ^{2} + y ^{2}}

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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