Question

Explain the difference between two antiderivatives of the function.

Short answer

The two functions are antiderivatives of each other if their difference is constant.

Step-by-step solution

Step1. Given

The two given functions are$f\left(x\right)\mathrm{and}g\left(x\right)$.

Step 2. Explanation

$$\begin{gathered} \mathrm{Any}\mathrm{two}\mathrm{functions}\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{derivative}\mathrm{must}\mathrm{differ}\mathrm{by}\mathrm{a}\mathrm{constant}. \\ \mathrm{Algebraically},\mathrm{this}\mathrm{means}\mathrm{that}\mathrm{to}\mathrm{find}\mathrm{one}\mathrm{antiderivative}\mathrm{of}\mathrm{a}\mathrm{function}, \\ \mathrm{then}\mathrm{all}\mathrm{other}\mathrm{antiderivatives}\mathrm{of}\mathrm{that}\mathrm{function}\mathrm{differ}\mathrm{from}\mathrm{the}\mathrm{one}\mathrm{that}\mathrm{find}\mathrm{by}\mathrm{a}\mathrm{constant}. \\ \mathrm{Explanation} \\ \mathrm{The}\mathrm{given}\mathrm{functions}\mathrm{g}\left(\mathrm{x}\right)\mathrm{and}\mathrm{h}\left(\mathrm{x}\right)\mathrm{are}\mathrm{both}\mathrm{antiderivatives}\mathrm{of}\mathrm{some}\mathrm{function}f\left(x\right). \\ \mathrm{Rewrite}\mathrm{the}\mathrm{functions}\mathrm{as},g\text{'}\left(x\right)=f\left(x\right)andh\text{'}\left(x\right)=f\left(x\right). \\ \mathrm{The}\mathrm{difference}\mathrm{of}\mathrm{these}\mathrm{functions}\mathrm{is},g\text{'}\left(x\right)-h\text{'}\left(x\right)=0 \\ T\mathrm{his}\mathrm{is}\mathrm{by}\mathrm{the}\mathrm{difference}\mathrm{rule}\mathrm{means}\mathrm{that},\left(8\right(x)-h(x\left)\right)=0 \\ \mathrm{Use}\mathrm{the}\mathrm{result},"\mathrm{If}\mathrm{is}\mathrm{zero}\mathrm{in}\mathrm{the}\mathrm{interior}\mathrm{of}/\mathrm{then}\mathrm{is}\mathrm{constant}\mathrm{on}/. \\ \mathrm{Form}\mathrm{the}\mathrm{resul}t, \\ \mathrm{if}g\text{'}\left(x\right)-\left(x\right)\mathrm{is}\mathrm{zero}\mathrm{then}g\left(x\right)-h\left(x\right)iscons\tan t. \\ \mathrm{Therefore},g\left(x\right)-h\left(x\right)=C\mathrm{for}\mathrm{some}\mathrm{real}\mathrm{number}C. \end{gathered}$$