Question

根据导数的定义求下列函数的导数：⑴求函数y=x²+3在x=1处的导数⑵求函数y=1/x在x=a(a≠0)处的导数

Answer 1

第一题：

y = x² + 3

y'|(x=1)

= lim(Δx→0) [f(1 + Δx) - f(1)]/Δx

= lim(Δx→0) {[(1 + Δx)² + 3] - 4}/Δx

= lim(Δx→0) {[1 + 2Δx + (Δx)² + 3] - 4}/Δx

= lim(Δx→0) [2Δx + (Δx)²]/Δx

= lim(Δx→0) (2 + Δx)

= 2

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y = 1/x

y'|(x=a)

= lim(Δx→0) [f(a + Δx) - f(a)]/Δx

= lim(Δx→0) [1/(a + Δx) - 1/a]/Δx

= lim(Δx→0) [a - (a + Δx)]/[aΔx(a + Δx)]

= lim(Δx→0) -Δx/[aΔx(a + Δx)]

= lim(Δx→0) -1/[a(a + Δx)]

= -1/[a(a + 0)]

= -1/a²