1. `y = 1/sqrt(1-2x)`;
2. `y = x/root 3 (1+x)`;
3. `y = x^2/(1-x)`;
4. `y = 1/(x(1-x))`;
5. `y = (2x+1)/(x^2-3x+2)`;
6. `y = x/(x^2-1)^2`;
7. `y = sin ax cos bx`;
8. `y = sin^6 x + cos^6 x`;
9. `y = sinh ax cos bx`;
10. `y = cosh ax`.

1. `y^((n)) = (2n-1)!!(1-2x)^(-1/2-n)`.
2. `y^((n)) = ((1+x)^(-1/3))^((n)) x + n ((1+x)^(-1/3))^((n-1))` `= (prod_(k=0)^(n-2) (1+3k)) / (-3)^(n-1) (-(x(3n-2))/(3(1+x)^(n+1/3)) + n/(1+x)^(n-2/3))`.
3. `y^((n)) = (1/(1-x))^((n)) x^2 + n(1/(1-x))^((n-1)) 2x + (n(n-1))/2 (1/(1-x))^((n-2)) * 2` `= n! (1-x)^(-n-1) x^2 + n! (1-x)^-n * 2x + n!(1-x)^(-n+1)` `= (n!)/(1-x)^n (x^2/(1-x) + x + 1)`.
4. `y^((n)) = (1/x + 1/(1-x))^((n))` `= (-1)^n n! x^(-n-1) + n! (1-x)^(-n-1)`.
5. `y^((n)) = (3/(x-2) - 2/(x-1))^((n))` `= (-1)^n n! [3(x-2)^(-n-1) - 2 (x-1)^(-n-1)]`.
6. `y^((n)) = 1/4(1/(x-1)^2 - 1/(x+1)^2)^((n))` `= 1/4(-1)^n (n+1)! [(x-1)^(-n-2) - (x+1)^(-n-2)]`.
7. 积化和差: `y^((n)) = 1/2(sin(a+b)x + sin(a-b)x)^((n))` `= 1/2[(a+b)^n sin((a+b)x + (n pi)/2) + (a-b)^n sin((a-b)x + (n pi)/2)]`.
8. `y = sin^6 x + (1-sin^2 x)^3` `= sin^6 x + 1 - 3sin^2 x + 3sin^4 x - sin^6 x` `= 1-3 sin^2 x cos^2 x` `= 1-3/4 sin^2 2x` `= 1-3/8 (1-cos 4x)` `= 5/8 + 3/8 cos 4x`. 所以 `y^((n)) = 3/8 * 4^n cos(4x + (n pi)/2)`.
9. 利用第 2 题的结果, `y^((n)) = (a^2+b^2)^(n/2) sinh ax cos (bx + n arctan {:b/a:})`.
10. n 为奇数时, `y^((n)) = a^n sinh ax`; n 为偶数时, `y^((n)) = a^n cosh ax`.

只证明第一个等式. 当 `n = 1` 时, 左边等于 `ae^(ax) sin(bx+c) + be^(ax) cos(bx+c)`
`= sqrt(a^2+b^2) e^(ax) sin(bx+c+theta)`
等于右边. 设等式对 `n-1` 成立, 则对正整数 `n`, 左边等于 `((a^2+b^2)^((n-1)/2) e^(ax) sin(bx+c+(n-1)theta))'`
`= (a^2+b^2)^((n-1)/2) e^(ax)`
`= a sin(bx+c+(n-1)theta) + b cos(bx+c + (n-1)theta)` 等于右边 □

用归纳法易证.

由 Leibniz 公式, `f^((n-1))(x)` `= sum_(k=0)^(n-1) (n-1;k) n!/(n-k)! (x-a)^(n-k) varphi^((n-1-k))(x)`,
`f^((n-1))(a) = 0`. 故 `f^((n))(a)` `= lim_(x to a) (f^((n-1))(x) - f^((n-1))(a))/(x-a)` `= lim_(x to a) sum_(k=0)^(n-1) (n-1;k) n!/(n-k)! (x-a)^(n-1-k) varphi^((n-1-k))(x)` `= n! varphi(x)`.

1. `y' = 1 + tan^2 x`,
   `y'' = 2 tan x + 2 tan^3 x`,
   `y''' = 2 + 8 tan^2 x + 6 tan^4 x`;
2. 一般地, 有 `y^((2n-1)) = sum_(k=0)^n a_(nk) tan^(2k) x`,
   `y^((2n)) = sum_(k=1)^n [2(k-1)a_(n\ k-1) + 2ka_(nk)] tan^(2k-1) x + 2na_(nn) tan^(2n+1) x`. 其中 `a_(nk)` 由以下递推公式确定: `a_(n+1, 0) = 2a_(n,1)`, `quad a_(n+1, n+1) = 2n(2n-1)a_(n,n)`, `a_(n+1, k) = 2(k-1)(2k-1)a_(n,k-1) + 8k^2 a_(n,k)` `+ 2(k+1)(2k+1)a_(n, k+1)`, `1 le k le n`.

1. `y = x^3 e^(2x)`;
2. `y = x^2 ln x`;
3. `y = sinh ax cos bx`;
4. `y = cosh ax sin bx`;
5. `y = e^x ln x`;
6. `y = ln x sin x`;
7. `y = e^x/x`;
8. `y = sin x / x^2`;
9. `y = ln^2 x`;
10. `y = (arctan x)^2`.

1. 对 `y = x^(n-1) ln x`, 有 `y^((n)) = ((n-1)!)/x`;
2. 对 `y = x^(n-1) e^(1/x)`, 有 `y^((n)) = (-1)^n/x^(n+1) e^(1/x)`.

1. `(1-x^2)y' - xy = 1`;
2. 成立递推公式: `y^((n)) = ((2n-1)xy^((n-1)) + (n-1)^2 y^((n-2))) / (1-x^2)` `(n ge 2)`;
3. 推导求函数 `z = (arcsin x)^2` 的高阶导数的递推公式.

1. `y -1/2 sin y = x`;
2. `y^2 + 2 ln y = x`;
3. `x^3 + y^3 -3axy = 0` `(a gt 0)`;
4. `x^4 + y^4 = a^4` `(a gt 0)`;
5. `sqrt(x^2+y^2) = ae^(arctan {:y/x:})` `(a gt 0)`;
6. `root 3 (x^2) + root 3 (y^2) = root 3 (a^2)` `(a gt 0)`.

1. `y = x + ln x` `(x gt 0)`;
2. `y = x + e^x`;
3. `y = 2x^2 - 4x^4`;
4. `y = 1/(1+x^2)`.

1. `x = 2t-t^2`, `y = 3t-t^3`;
2. `x = a cos^3 t`, `y = at sin^3 t`;
3. `x = e^t cos t`, `y = e^t sin t`;
4. `x = arcsin t/sqrt(1+t^2)`, `y = arccos 1/sqrt(1+t^2)`.

1. `y = (x^3+2x) sin^2 x`;
2. `y = e^(2x) ln x` `(x gt 0)`;
3. `y = sqrt (1+x^2)`;
4. `y = x^x` `(x gt 0)`.