derivative of y^x

解答

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

(ln (y) + \frac{x \frac{d y}{d x}}{y}) y^{x}

求解步骤

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

使用指数法则:

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

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

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

使用连锁律:

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

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

\frac{d}{d x} (x ln (y)) = ln (y) + \frac{x \frac{d y}{d x}}{y}

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

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

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

d er i v a t i v e f (x) = 7 x + 6

\int \frac{1}{2 tan ( x )} d x

in v erse l a pl a ce \frac{s ^{2} + 3 s + 5}{s ^{3} ( s + 2 )}

\frac{d y}{d x} = x

n = 0 \sum \infty \frac{2}{n ^{2}}