derivative of ln(x^2+1-e^{x/2}cos(pix))

解答

\frac{d}{d x} (ln (x^{2} + 1) - e^{\frac{x}{2}} cos (π x))

\frac{2 x}{x ^{2} + 1} - \frac{1}{2} e^{\frac{x}{2}} cos (π x) + π e^{\frac{x}{2}} sin (π x)

求解步骤

\frac{d}{d x} (ln (x^{2} + 1) - e^{\frac{x}{2}} cos (π x))

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (ln (x^{2} + 1)) - \frac{d}{d x} (e^{\frac{x}{2}} cos (π x))

\frac{d}{d x} (ln (x^{2} + 1)) = \frac{2 x}{x ^{2} + 1}

\frac{d}{d x} (e^{\frac{x}{2}} cos (π x)) = \frac{1}{2} e^{\frac{x}{2}} cos (π x) - π e^{\frac{x}{2}} sin (π x)

= \frac{2 x}{x ^{2} + 1} - (\frac{1}{2} e^{\frac{x}{2}} cos (π x) - π e^{\frac{x}{2}} sin (π x))

化简

= \frac{2 x}{x ^{2} + 1} - \frac{1}{2} e^{\frac{x}{2}} cos (π x) + π e^{\frac{x}{2}} sin (π x)

\int \frac{4 x + 5}{x ^{2} + 4 x + 2} d x

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

\frac{d y}{d x} = (\frac{( 2 y + 3 )}{( 4 x + 5 )})^{2}

a re a cos (6 x), 0

y^{^{''}} + 43 y^{^{'}} + 12 y = 0, y (0) = 1, y^{^{'}} (0) = 0