求解:用导数的基本定义来证明下列公式1.(x^a)'=ax^(a-1)2.(a^x)'=a^xlna3.(logax)'=1/xlna4.(lnx)'=1/x

Question

求解:用导数的基本定义来证明下列公式
1.(x^a)'=ax^(a-1)
2.(a^x)'=a^xlna
3.(logax)'=1/xlna
4.(lnx)'=1/x

Answer 1

1、a≠0时,(x^a)'
=lim(h→0)((x+h)^a-x^a)/h
=lim(h→0)x^a*((1+h/x)^a-1)/h
=x^a*lim(h→0)(e^(aln(1+h/x))-1)/h
=x^a*lim(h→0)(e^(aln(1+h/x))-1)/(aln(1+h/x))*aln(1+h/x)/(h/x)*(h/x)/h
=x^a*1*a*1/x
=ax^(a-1)
2、(a^x)'
=lim(h→0)(a^(x+h)-a^x)/h
=lim(h→0)a^x*(a^h-1)/h
=a^x*lim(h→0)(e^(hlna)-1)/(hlna)*hlna/h
=a^x*1*lna
=a^xlns
证明3之前先证明4吧.
4、(lnx)'
=lim(h→0)(ln(x+h)-lnx)/h
=lim(h→0)ln(1+h/x)/(h/x)*(h/x)/h
=1*1/x
=1/x
3、(logax)'
=(lnx/lna)'
=(lnx)'/lna
=(1/x)/lna
=1/(xlna)