3.2.2 导数的运算法则（学案）-【成才之路】2020-2021学年高中新课程数学同步学习指导（人教A版选修1-1）

现行旧教材·高中新课程学习指导 　 　 跟踪练习 1:(1)y′ = 1 x2( )′ = (x - 2 )′ = - 2x - 3 . (2)y′ = (3 x)′ = (x 1 3 )′ = 1 3 x - 2 3 . (3)y′ = (2x)′ = 2xln 2. (4)y′ = (log 3 x)′ = 1 xln 3 . 　 　 典例试做 2:f ′(x) = 1 x( )′ = (x - 12 )′ = - 1 2 x - 1 2 - 1 = - 1 2 x - 3 2 = - 1 2 x3 , ∴ f ′(1) = - 1 2 1 = - 1 2 , ∴ 函数 f(x)在 x = 1 处的导数为 - 1 2 . 　 　 跟踪练习 2:f ′(x) = 1 n x( )′ = (x - 1 n )′ = - 1 n x - 1 n - 1 = - 1 n x - n + 1 n , ∴ f ′(1) = - 1 n , 由 f ′(1) = - 1 3 得 - 1 n = - 1 3 ,得 n = 3. 　 　 典例试做 3:∵ y = cos x,∴ y′ = - sin x, 曲线在点 P π3 , 1 2( )处的切线斜率是 y′ | x = π3 = - sin π 3 = - 3 2 . ∴ 过点 P 且与切线垂直的直线的斜率为 2 3 , ∴ 所求的直线方程为 y - 1 2 = 2 3 x - π 3( ), 即 2x - 3y - 2π 3 + 3 2 = 0. 　 　 跟踪练习 3:y = x - 1　 由 y = lnx 得 y′ = 1 x ,令 x = 1 得 y′ = 1 即切线斜率为 1,∴ 切线方程为 y = x -1. 　 　 典例试做 4:由于点 B(3,5) 不在曲线上,所以点 B 不是切 点,设切点坐标为(x0 ,y0 ). ∵ y = x2 ,∴ y′ = 2x, ∴ 切线斜率为 k = 2x0 , ∴ 切线方程为:y - x20 = 2x0 (x - x0 ). ∵ B(3,5)在切线上, ∴ 5 - x20 = 2x0 (3 - x0 ), 解之,得 x0 = 1 或 x0 = 5. 所以所求切线方程为 y - 1 = 2(x - 1)或 y - 25 = 10(x - 5), 即 2x - y - 1 = 0 或 10x - y - 25 = 0. 　 　 跟踪练习 4:由 f(x) = 2f(2 - x) - x2 + 8x - 8,令(2 - x) 取 代 x,得 f(2 - x) = 2f(x) - (2 - x)2 + 8(2 - x) - 8, 即 2f(x) - f(2 - x) = x2 + 4x - 4, 联立 f(x) = 2f(2 - x) - x2 + 8x - 8,得f(x) = x2 , ∴ f ′(x) = 2x,f ′(2) = 4,即所求切线斜率为 4, ∴ 切线方程为 y - 4 = 4(x - 2),即 4x - y - 4 = 0. 　 　 典例试做 5:∵ y′ = (2x)′ = 2xln 2,∴ y′ | x = 1 = 2ln 2, 又 x = 1 时,y = 2,∴ 切线方程为 y - 2 = 2ln2 (x - 1),即 2xln 2 - y - 2ln 2 + 2 = 0. 课堂达标·固基础 1. A　 2. D　 3. D 4. 1 8 5. 由题图可知曲线的切线 l 经过点(1,2),则 k + 3 = 2,得 k = - 1, 即f ′(1) = - 1. 3. 2. 2　 导数的运算法则 新知导学 　 　 f′(x) ± g′(x)　 f′(x)g(x) + f(x)·g′(x) f′(x)g(x) - f(x)g′(x) g2 (x) 预习自测 1. A　 ∵ f(x) = ax2 + c,∴ f ′(x) = 2ax, 又∵ f ′(1) = 2a,∴ 2a = 2,∴ a = 1. 2. C　 f′(x) = (ex)′ln x + ex(ln x)′ = exln x + e x x = e x(xln x + 1) x . 3. B　 ∵ f(x) = x4 - 2x3 ,∴ f′(x) = 4x3 - 6x2 ,∴ f′(1) = - 2, 又 f(1) = 1 - 2 = - 1, ∴ 所求的切线方程为 y + 1 = - 2(x - 1),即 y = - 2x + 1. 故 选 B. 4. 1　 由于 f′(x) = e x(x + a) - ex (x + a)2 ,故 f′(1) = ea (1 + a)2 = e 4 ,解得 a = 1. 5. (1)y′ = (sin x - 2x2