derivative f(x)=x^pi*pi^x

解答

导数

f (x) = x^{π} \cdot π^{x}

π^{1 + x} x^{π - 1} + ln (π) π^{x} x^{π}

求解步骤

\frac{d}{d x} (x^{π} π^{x})

使用微分乘法定则:

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

f = x^{π}, g = π^{x}

= \frac{d}{d x} (x^{π}) π^{x} + \frac{d}{d x} (π^{x}) x^{π}

\frac{d}{d x} (x^{π}) = π x^{π - 1}

\frac{d}{d x} (π^{x}) = π^{x} ln (π)

= π x^{π - 1} π^{x} + π^{x} ln (π) x^{π}

使用指数法则:

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

π π^{x} = π^{1 + x}

= π^{1 + x} x^{π - 1} + ln (π) π^{x} x^{π}

\int 12 x^{2} sin (6 - x^{3}) d x

\int \frac{x}{2 x ^{2} - x + 1} d x

\int_{- 1}^{1} \frac{1}{x + 1} d x

\int x^{3} + x d x

in v erse l a pl a ce \frac{1}{s} \cdot (e^{- 2 \cdot s \cdot π} - 1)