Question

四个函数求导，求详解

Answer 1

f1(x) = e^[sin(x)][x^2 + x +3]^(3/2),
df1(x)/dx = e^[sin(x)]cos(x)[x^2 + x + 3]^(3/2) + e^[sin(x)](3/2)[x^2 + x + 3]^(1/2)(2x + 1)
= e^[sin(x)][x^2 + x + 3]^(1/2)[cos(x)(x^2 + x + 3) + (3/2)(2x+1)]

f2(x) = ln[1+x]^(1/2) - ln(1-x) = (1/2)ln(1+x) - ln(1-x),
df2(x)/dx = (1/2)*1/(1+x) - (-1)/(1-x) = 1/[2(1+x)] + 1/(1-x).

f3(x) = ln[tan(x/2)],
df3(x)/dx = 1/tan(x/2)*[sec(x/2)]^2*(1/2) = [sec(x/2)]^2/[2tan(x/2)] = 1/[2sin(x/2)cos(x/2)] = 1/sin(x).

f4(x) = ln[x + (1+x^2)^(1/2)]
df4(x)/dx = 1/[x + (1+x^2)^(1/2)]*[1 + (1/2)(1+x^2)^(-1/2)*(2x)] = 1/[x + (1+x^2)^(1/2)]*[ 1 + x/(1+x^2)^(1/2)]
= 1/[x + (1+x^2)^(1/2)]*[x + (1+x^2)^(1/2)]/(1+x^2)^(1/2)
= 1/(1+x^2)^(1/2)