(d^4)/(dx^4)(x^x)

解

\frac{d ^{4}}{d x ^{4}} (x^{x})

x^{x} (ln (x) + 1)^{4} + 3 x^{x - 1} (ln (x) + 1)^{2} + 3 x^{x - 1} ln^{2} (x) + 6 x^{x - 1} ln (x) - 4 x^{x - 2} ln (x) - x^{x - 2} + 3 x^{x - 1} + 2 x^{x - 3}

解答ステップ

\frac{d ^{4}}{d x ^{4}} (x^{x})

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

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

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

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

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

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

\frac{d}{d x} (x^{x} (ln (x) + 1)^{3} + 3 x^{x - 1} ln (x) + 3 x^{x - 1} - x^{x - 2}) = x^{x} (ln (x) + 1)^{4} + 3 x^{x - 1} (ln (x) + 1)^{2} + 3 x^{x - 1} ln^{2} (x) + 6 x^{x - 1} ln (x) - 4 x^{x - 2} ln (x) - x^{x - 2} + 3 x^{x - 1} + 2 x^{x - 3}

= x^{x} (ln (x) + 1)^{4} + 3 x^{x - 1} (ln (x) + 1)^{2} + 3 x^{x - 1} ln^{2} (x) + 6 x^{x - 1} ln (x) - 4 x^{x - 2} ln (x) - x^{x - 2} + 3 x^{x - 1} + 2 x^{x - 3}