Question

If $f\left(x\right)=x+1$, then write the value of $\frac{d}{dx}\left(fof\right)\left(x\right)$.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},f\left(x\right)=x+1 \\ \mathrm{Now},\left(fof\right)\left(x\right)=f\left(f\left(x\right)\right) \\ \Rightarrow \left(fof\right)\left(x\right)=f\left(x+1\right) \\ \Rightarrow \left(fof\right)\left(x\right)=\left(x+1\right)+1 \\ \Rightarrow \left(fof\right)=x+2 \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{d}{dx}\left\{\left(fof\right)\left(x\right)\right\}=\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(2\right) \\ \Rightarrow \frac{d}{dx}\left\{\left(fof\right)\left(x\right)\right\}=1+0 \\ \Rightarrow \frac{d}{dx}\left\{\left(fof\right)\left(x\right)\right\}=1 \end{gathered}$$