Question 6.8.4: Find x'(t) when x(t) = 5(1 +√t³ + 1)^25....

The initial step is easy: let x(t)=5u^{25},{\mathrm{where}}\;u=1+{\sqrt{t^{3}+1}}, to obtain

x^{\prime}(t)=5\cdot25u^{24}{\frac{\mathrm{d}u}{\mathrm{d}t}}=125u^{24}{\frac{\mathrm{d}u}{\mathrm{d}t}}   (∗)

The new feature in this example is that we cannot write down du/dt at once. Finding du/dt requires using the chain rule a second time. Let u=1+{\sqrt{v}}=1+v^{1/2},{\mathrm{where}}\,v=t^{3}+1. Then

{\frac{\mathrm{d}u}{\mathrm{d}t}}={\frac{1}{2}}v^{(1/2)-1}\cdot{\frac{\mathrm{d}v}{\mathrm{d}t}}={\frac{1}{2}}v^{-1/2}\cdot3t^{2}={\frac{1}{2}}(t^{3}+1)^{-1/2}\cdot3t^{2}     (∗∗)

From (∗) and (∗∗), we get