微积分(A)随缘一题[8]

\[\frac{dx}{dy}=\frac{1}{f'(x)} \]

\[\frac{d\frac{dx}{dy}}{dy}=\frac{d \frac{1}{f'(x)}}{dx} \cdot \frac{1}{\frac{dy}{dx}}=\frac{f''(x)\frac{-1}{[f'(x)]^2}}{f'(x)}=-\frac{f''(x)}{[f'(x)]^3} \]

\[\frac{d \frac{dx^2}{d^2y}}{dy}=\frac{d \frac{dx^2}{d^2y}}{dx} \cdot \frac{1}{\frac{dy}{dx}}=\frac{-\frac{f'''(x)[f'(x)]^3-f''(x) \cdot f''(x) \cdot 3[f'(x)]^2}{[f'(x)]^6}}{f'(x)}=\frac{3[f'(x) \cdot f''(x)]^2-f'''(x)[f'(x)]^3}{[f'(x)]^7} \]