Question

求导数的题

求导数的题第3题求大神写下步骤

Answer 1

有真数分子的二次根式知x≥-½→分母三次根式>0→定义域x>0
原函数=lnx+½ln(2x+1)-⅓ln(3x+1)
y'=1/x+½·2/(2x+1)-⅓·3/(3x+1)=1/x+1/(2x+1)-1/(3x+1)
y'(1)=1+⅓-¼=13/12

Answer 2

(1) ∫(ln2,ln3)1/[e^x-e^(-x)]dx
=∫(ln2,ln3) e^x/[(e^x)^2-1]dx
=∫(ln2,ln3) 1/[(e^x)^2-1]de^x
令t=e^x
t1=e^ln2=2
t2=e^ln3=3
上式=∫(2,3)1/(t^2-1)dt
=1/2∫(2,3)[1/(t-1)-1/(t+1)]dt
=1/2ln[(t-1)/(t+1)]|(2,3)
=1/2{ln[(3-1)/(3+1)]-ln[(2-1)/(2+1)]}
=1/2[ln(1/2)-ln(1/3)]
=1/2ln[(1/2)/(1/3)]
=1/2ln(3/2)