derivative of f(xx^'(x))

解答

\frac{d}{d x} (f (x) x^{^{'}} (x))

(' + 1) x^{'} f (x) + \frac{d}{d x} (f (x)) x^{' + 1}

求解步骤

\frac{d}{d x} (f (x) x^{^{'}} (x))

化简

f (x) x^{^{'}} (x) : x^{^{'} + 1} f (x)

= \frac{d}{d x} (x^{^{^{'}} + 1} f (x))

使用微分乘法定则:

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

f = x^{^{'} + 1}, g = f (x)

= \frac{d}{d x} (x^{^{^{'}} + 1}) f (x) + \frac{d}{d x} (f (x)) x^{^{^{'}} + 1}

\frac{d}{d x} (x^{^{'} + 1}) = (^{'} + 1) x^{^{'}}

= (^{^{'}} + 1) x^{^{^{'}}} f (x) + \frac{d}{d x} (f (x)) x^{^{^{'}} + 1}

\frac{\partial}{\partial x} (ln (8 x^{2} + 3 y^{2} + 3))

\int \frac{10 x ^{4} - 9 x ^{2}}{2 x ^{5} - 3 x ^{3} + 14} d x

\frac{\partial}{\partial x} (\frac{5 y}{x ^{3}})

y^{^{''}} - 9 y^{^{'}} + 18 y = 5 sin (4 t), y (0) = 5, y^{^{'}} (0) = 2

t a y l or x e^{- 3 x}