Page:Calculus Made Easy.pdf/41

 Doing the cubing we obtain

Now we know that we may neglect small quantities of the second and third orders; since, when $$dy$$ and $$dx$$ are both made indefinitely small, $$(dx)^2$$ and $$(dx)^3$$ will become indefinitely smaller by comparison. So, regarding them as negligible, we have left:

But $$y=x^3$$; and, subtracting this, we have:

and

Case 3.

Try differentiating $$y=x^4$$. Starting as before by letting both $$y$$ and $$x$$ grow a bit, we have:

Working out the raising to the fourth power, we get

Then striking out the terms containing all the higher powers of $$dx$$, as being negligible by comparison, we have

Subtracting the original $$y=x^4$$, we have left

and