derivative of (x^{{p}(x})/(x^m-a^m))

解答

\frac{d}{d x} (\frac{x ^{p (x)}}{x ^{m} - a ^{m}})

\frac{x ^{p (x)} ( ln ( x ) \frac{d}{d x} ( p ( x ) ) + \frac{p ( x )}{x} ) ( x ^{m} - a ^{m} ) - m x ^{m - 1 + p (x)}}{( x ^{m} - a ^{m} ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{x ^{p (x)}}{x ^{m} - a ^{m}})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( x ^{p (x)} ) ( x ^{m} - a ^{m} ) - \frac{d}{d x} ( x ^{m} - a ^{m} ) x ^{p (x)}}{( x ^{m} - a ^{m} ) ^{2}}

\frac{d}{d x} (x^{p (x)}) = x^{p (x)} (ln (x) \frac{d}{d x} (p (x)) + \frac{p ( x )}{x})

\frac{d}{d x} (x^{m} - a^{m}) = m x^{m - 1}

= \frac{x ^{p (x)} ( ln ( x ) \frac{d}{d x} ( p ( x ) ) + \frac{p ( x )}{x} ) ( x ^{m} - a ^{m} ) - m x ^{m - 1} x ^{p (x)}}{( x ^{m} - a ^{m} ) ^{2}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

x^{m - 1} x^{p (x)} = x^{m - 1 + p (x)}

= \frac{x ^{p (x)} ( ln ( x ) \frac{d}{d x} ( p ( x ) ) + \frac{p ( x )}{x} ) ( x ^{m} - a ^{m} ) - m x ^{m - 1 + p (x)}}{( x ^{m} - a ^{m} ) ^{2}}

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

\frac{d}{d x} (\frac{1 + csc ( x ^{7} )}{1 - cot ( x ^{7} )})

d er i v a t i v e \frac{9}{x - 6}

\int_{1}^{5} 1 + (4)^{2} d x

\frac{d}{d x} (\frac{3}{2 x})