Page:Calculus Made Easy.pdf/38



Now let us see how, on first principles, we can differentiate some simple algebraical expression.

Case 1.

Let us begin with the simple expression $$y=x^2$$. Now remember that the fundamental notion about the calculus is the idea of growing. Mathematicians call it varying. Now as $$y$$ and $$x^2$$ are equal to one another, it is clear that if $$x$$ grows, $$x^2$$ will also grow. And if $$x^2$$ grows, then $$y$$ will also grow. What we have got to find out is the proportion between the growing of $$y$$ and the growing of $$x$$. In other words our task is to find out the ratio between $$dy$$ and $$dx$$, or, in brief, to find the value of $$\frac{dy}{dx}$$.

Let $$x$$, then, grow a little bit bigger and become $$x+dx$$; similarly, $$y$$ will grow a bit bigger and will become $$y+dy$$. Then, clearly, it will still be true that the enlarged $$y$$ will be equal to the square of the enlarged $$x$$. Writing this down, we have: $y+dy=(x+dx)^2$. Doing the squaring we get: $y+dy=x^2+2x \cdot dx+(dx)^2$.