derivative of (sqrt(x^2+1)(x^3+6))

解答

\frac{d}{d x} ((x^{2} + 1) (x^{3} + 6))

\frac{4 x ^{4} + 3 x ^{2} + 6 x}{x ^{2} + 1}

求解步骤

\frac{d}{d x} ((x^{2} + 1) (x^{3} + 6))

使用微分乘法定则:

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

f = x^{2} + 1, g = x^{3} + 6

= \frac{d}{d x} (x^{2} + 1) (x^{3} + 6) + \frac{d}{d x} (x^{3} + 6) x^{2} + 1

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

\frac{d}{d x} (x^{3} + 6) = 3 x^{2}

= \frac{x}{x ^{2} + 1} (x^{3} + 6) + 3 x^{2} x^{2} + 1

化简

\frac{x}{x ^{2} + 1} (x^{3} + 6) + 3 x^{2} x^{2} + 1 : \frac{4 x ^{4} + 3 x ^{2} + 6 x}{x ^{2} + 1}

= \frac{4 x ^{4} + 3 x ^{2} + 6 x}{x ^{2} + 1}

l a pl a ce t r an s f or m e^{t} sin (3 t)

d er i v a t i v e 3 x^{2} + x y + 3 y^{2}

\frac{d}{d x} (\frac{x ^{2}}{100} + 8 + \frac{20}{x})

\frac{d}{d x} (cos (x) - 6 tan (x))

\frac{d}{d x} (\frac{3 x ^{2} + 9}{6 x})