Answer
a) $dy/dx=1$
b) $dy/dx=1$
Work Step by Step
a) $y=(u/5)+7$ and $u=5x-35$
We have $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}=\Big(\frac{u}{5}+7\Big)'(5x-35)'=\frac{1}{5}\times5=1$$
b) $y=1+(1/u)$ and $u=1/(x-1)$
We have $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}=\Big(1+\frac{1}{u}\Big)'\Big(\frac{1}{x-1}\Big)'=\Big(\frac{-1(u)'}{u^2}\Big)\Big(\frac{-1(x-1)'}{(x-1)^2}\Big)$$
$$\frac{dy}{dx}=\Big(-\frac{1}{u^2}\Big)\Big(-\frac{1}{(x-1)^2}\Big)=\frac{1}{u^2(x-1)^2}$$
Here we substitute $u=1/(x-1)$: $$\frac{dy}{dx}=\frac{1}{\frac{1}{(x-1)^2}\times(x-1)^2}=\frac{1}{1}=1$$
In both cases, we see that the results are the same and equal with $dy/dx$ when $y=x$. So writing the function as a composite in different ways does not change the value of its derivative.
