Page:Calculus Made Easy.pdf/211



is the process by which when $$y$$ is given us (as a function of $$x$$), we can find $$\frac {dy}{dx}$$.

Like every other mathematical operation, the process of differentiation may be reversed; thus, if differentiating $$y=x^4$$ gives us $$\frac {dy}{dx}=4x^3$$; if one begins with $$\frac {dy}{dx}=4x^3$$ one would say that reversing the process would yield $$y=x^4$$. But here comes in a curious point. We should get $$\frac {dy}{dx}=4x^3$$ if we had begun with any of the following: $$x^4$$, or $$x^4+a$$, or $$x^4+c$$, or $$x^4$$ with any added constant. So it is clear that in working backwards from $$\frac {dy}{dx}$$ to $$y$$, one must make provision for the possibility of there being an added constant, the value of which will be undetermined