derivative f(x)=(x+1)^{2x}

解答

导数

f (x) = (x + 1)^{2 x}

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

求解步骤

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

使用指数法则:

a^{b} = e^{b l n (a)}

(x + 1)^{2 x} = e^{2 x l n (x + 1)}

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

使用连锁律:

e^{2 x l n (x + 1)} \frac{d}{d x} (2 x ln (x + 1))

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

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

= e^{2 x l n (x + 1)} \cdot 2 (ln (x + 1) + \frac{x}{x + 1})

e^{2 x l n (x + 1)} = (x + 1)^{2 x}

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

roo t s \frac{x}{a}

t an g e n t \frac{x}{y} = sin (\frac{π x y}{2}), (1, 1)

\frac{\partial}{\partial x} (y^{2} + 2 x y - x^{2})

x \to 16 lim (x + 2 x)

d er i v a t i v e y = x \cdot x