高等数学_例题解-4

例4求函数(x)=a(a>0,a≠=1)的导数 MF /(x)=im / (x+h-f(x)_lim arth-a' h2→>0h h->0 令 =ana oge 即(a) aIn a 特别地(e)y=ex

结束 例4 求函数f(x)=a x (a0 a1) 的导数 解 解 f (x) h f x h f x h ( ) ( ) lim 0 + − = → h a a x h x h − = + →0 lim h a a h h x 1 lim 0 − = → a t h − = ====== 令 1 log (1 ) lim 0 t t a a t x → + a a e a x a x ln log 1 = =  即 (a x )=a x ln a 特别地(e x )=e x  解 f (x) h f x h f x h ( ) ( ) lim 0 + − = → h a a x h x h − = + →0 lim h a a h h x 1 lim 0 − = → 解 f (x) h f x h f x h ( ) ( ) lim 0 + − = → h a a x h x h − = + →0 lim h a a h h x 1 lim 0 − = → a t h − = ====== 令 1 log (1 ) lim 0 t t a a t x → + a a e a x a x ln log 1 = =  a t h − = ====== 令 1 log (1 ) lim 0 t t a a t x → + a a e a x a x ln log 1 = = 