Simplest Cases

Excerpts

• Simplest Cases

  Now remember that the fundamental notion about the calculus is the idea of *growing*. Mathematicians call it *varying*.

• Simplest Cases

  Then $(dx)^2$ will mean a little bit of a little bit of $x$; that is, as explained above (smallness), it is a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms.

• Simplest Cases

  But, you will say, we neglected a whole unit.

• Simplest Cases

  Just look at these results: the operation of differentiating appears to have had the effect of diminishing the power of $x$ by $1$ (for example in the last case reducing $x^4$ to $x^3$), and at the same time multiplying by a number (the same number in fact which originally appeared as the power).

• Simplest Cases

  To differentiate $x^n$, multiply by the power and reduce the power by one, so giving us $nx^{n-1}$ as the result.

• Simplest Cases

  You have now learned how to differentiate powers of $x$. How easy it is!

• Simplest Cases

  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.

• Simplest Cases

  But, you will say, we neglected a whole unit. Well, try again, making $dx$ a still smaller bit.

Equations

• Simplest Cases

  \frac{dy}{dx} = 2x

  Differentiating y = x^2 with respect to x gives the ratio of the growing of y to the growing of x equal to 2x.

• Simplest Cases

  \frac{dy}{dx} = 3x^2

  Differentiating y = x^3 with respect to x gives dy/dx equal to 3x^2.

• Simplest Cases

  \frac{dy}{dx} = 4x^3

  Differentiating y = x^4 with respect to x gives dy/dx equal to 4x^3.

• Simplest Cases

  \frac{dy}{dx} = 5x^4

  Differentiating y = x^5 with respect to x gives dy/dx equal to 5x^4.

• Simplest Cases

  \frac{dy}{dx} = nx^{(n-1)}

  For y = x^n, the differential coefficient of y with respect to x is n times x raised to the power n minus one; the chapter states this as the general rule, holding for whole positive n and checked for negative and fractional n.

• Simplest Cases

  \frac{dy}{dx} = -2x^{-3}

  Differentiating y = x^(-2) with respect to x gives dy/dx equal to -2x^(-3), in agreement with the general power rule.

• Simplest Cases

  \dfrac{dy}{dx} = \dfrac{1}{2} x^{-\efrac{1}{2}}

  Differentiating y = x^(1/2) with respect to x gives dy/dx equal to one half times x^(-1/2), agreeing with the general power rule for a fractional power.

Problems