derivative of ({f}(x,h(x+h)-{f}(x))/h)

解答

\frac{d}{d x} (\frac{f ( x , h ) ( x + h ) - f ( x )}{h})

\frac{1}{h} ((\frac{d}{d x} (f (x, h)) (x + h) + 1 \cdot f (x, h)) - \frac{d}{d x} (f (x)))

求解步骤

\frac{d}{d x} (\frac{f ( x , h ) ( x + h ) - f ( x )}{h})

将

h

视为常数

将常数提出:

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

= \frac{1}{h} \frac{d}{d x} (f (x, h) (x + h) - f (x))

使用微分加减法定则:

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

= \frac{1}{h} (\frac{d}{d x} (f (x, h) (x + h)) - \frac{d}{d x} (f (x)))

\frac{d}{d x} (f (x, h) (x + h)) = \frac{d}{d x} (f (x, h)) (x + h) + 1 \cdot f (x, h)

= \frac{1}{h} ((\frac{d}{d x} (f (x, h)) (x + h) + 1 \cdot f (x, h)) - \frac{d}{d x} (f (x)))

\frac{d}{d n} (i^{n})

x \to 1 lim (\frac{x ^{3}}{3})

d er i v a t i v e f (x) = \frac{x}{5 + x}

t an g e n t f (x) = \frac{x}{x + 1}, at x = 4

l a pl a ce t r an s f or m 2 s