derivative of x^{x^{x^x}}

解答

\frac{d}{d x} (x^{x^{x^{x}}})

x^{x^{x^{x}}} (x^{x^{x}} ln (x) (x^{x} ln (x) (ln (x) + 1) + x^{x - 1}) + x^{x^{x} - 1})

求解步骤

\frac{d}{d x} (x^{x^{x^{x}}})

使用指数法则:

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

x^{x^{x^{x}}} = e^{x^{x^{x}} l n (x)}

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

使用连锁律:

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

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

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

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

e^{x^{x^{x}} l n (x)} = x^{x^{x^{x}}}

= x^{x^{x^{x}}} (x^{x^{x}} ln (x) (x^{x} ln (x) (ln (x) + 1) + x^{x - 1}) + x^{x^{x} - 1})

y^{^{''}} + 26 y = 0

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