Answer
$$\frac{d\lambda}{du}=\frac{6(au+b)^5(ad-bc)}{(cu+d)^7}$$
Work Step by Step
$$\frac{d\lambda}{du}=\frac{d}{du}\left(\left(\frac{au+b}{cu+d}\right)^6\right)=6\left(\frac{au+b}{cu+d}\right)^5\frac{d}{du}\left(\frac{au+b}{cu+d}\right)=
6\left(\frac{au+b}{cu+d}\right)^5\frac{\frac{d}{du}(au+b)(cu+d)-(au+b)\frac{d}{du}(cu+d)}{(cu+d)^2}=
6\left(\frac{au+b}{cu+d}\right)^5\frac{a(cu+d)-(au+b)\cdot c}{(cu+d)^2}=
6\left(\frac{au+b}{cu+d}\right)^5\frac{acu+ad-acu-bc}{(cu+d)^2}=
6\left(\frac{au+b}{cu+d}\right)^5\frac{ad-bc}{(cu+d)^2}=\frac{6(au+b)^5(ad-bc)}{(cu+d)^7}$$
