Page:Calculus Made Easy.pdf/43

 and subtracting $$y=x^5$$ leaves us

whence

exactly as we supposed.

Following out logically our observation, we should conclude that if we want to deal with any higher power,—call it $$n$$—we could tackle it in the same way.

Let

then, we should expect to find that

For example, let $$n=8$$, then $$y=x^8$$; and differentiating it would give $$\frac{dy}{dx}=8x^7$$.

And, indeed, the rule that differentiating $$x^n$$ gives as the result $$nx^{n-1}$$ is true for all cases where $$n$$ is a whole number and positive. [Expanding $$(x+dx)^n$$ by the binomial theorem will at once show this.] But the question whether it is true for cases where $$n$$ has negative or fractional values requires further consideration.

Case of a negative power.

Let $$y=x^{-2}$$. Then proceed as before: