Use a substitution to differentiate the expression with respect to x: (1)/(x + 1) + x​

Question

Use a substitution to differentiate the expression with respect to x:


`(1)/(x + 1) + x`​

Answer 1

`\bf y=\cfrac{1}{x+1}+x\qquad \qquad \begin{cases} u=x+1\\ (du)/(dx)=1\\[-0.5em] \hrulefill\\ u-1=x \end{cases}\qquad \qquad y=\cfrac{1}{u}+(u-1) \\\\\\ y=u^(-1)+u-1\implies \cfrac{dy}{du}=\stackrel{\textit{chain-rule}}{-u^(-2)\cdot \cfrac{du}{dx}}+\stackrel{\textit{chain-rule}}{1\cdot \cfrac{du}{dx}}+0 \\\\\\ \cfrac{dy}{dx}=-\cfrac{1}{u^2}\cdot 1+1\cdot 1\implies \cfrac{dy}{dx}=-\cfrac{1}{u^2}+1\implies \cfrac{dy}{dx}=-\cfrac{1}{(x+1)^2}+1`