derivative of a(x+1(x-{r}(x)))

解答

\frac{d}{d x} (a (x + 1) (x - r (x)))

a (2 x - x \frac{d}{d x} (r (x)) - r (x) - \frac{d}{d x} (r (x)) + 1)

求解步骤

\frac{d}{d x} (a (x + 1) (x - r (x)))

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= a \frac{d}{d x} ((x + 1) (x - r (x)))

使用微分乘法定则:

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

f = x + 1, g = x - r (x)

= a (\frac{d}{d x} (x + 1) (x - r (x)) + \frac{d}{d x} (x - r (x)) (x + 1))

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

\frac{d}{d x} (x - r (x)) = 1 - \frac{d}{d x} (r (x))

= a (1 \cdot (x - r (x)) + (1 - \frac{d}{d x} (r (x))) (x + 1))

化简

a (1 \cdot (x - r (x)) + (1 - \frac{d}{d x} (r (x))) (x + 1)) : a (2 x - x \frac{d}{d x} (r (x)) - r (x) - \frac{d}{d x} (r (x)) + 1)

= a (2 x - x \frac{d}{d x} (r (x)) - r (x) - \frac{d}{d x} (r (x)) + 1)

\int \frac{- 2 \cdot x ^{4}}{2 \cdot x ^{3} - 3 \cdot x ^{2}} d x

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

\frac{d y}{d x} + 6 y = 0

\int_{2}^{3} x^{5} d x

a re a f (x) = x^{2} - 2 x + 1, g (x) = - x + 3