I'm relearning calc using Sylvanus Thompson's text. Differentiation is taught, not by the rule, but by adding differentials (i.e dy, dx) and working out algebraically. I had this figured out, but can't remember: let Y=X^-2. Then he posits Y+dy=(X+dx)^-2. Next, he says this is equal to X^-2 (1+dx/x)^-2.....ok, this begininng of problem and I can't figure out what he's doing algebraically to get that last part. Any help would be appreciated
Calc questionon differentiation: let Y=X^-2; then Sylvanus posits Y+dy=(X+dx)^-2.
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\(\displaystyle (X + dx)^{-2}\)
\(\displaystyle \dfrac{1}{(X + dx)^{2}}\)
\(\displaystyle \dfrac{1}{X^2 + 2\cdot dx \cdot X + dx^2}\)
\(\displaystyle \dfrac{1}{X^2 + 2\cdot dx \cdot X(\frac{X}{X}) + dx^2(\frac{X^2}{X^2})}\)
\(\displaystyle \dfrac{1}{X^2 + \frac{2\cdot dx \cdot X^2}{X} + \frac{dx^2\cdot X^2}{X^2}}\)
\(\displaystyle \dfrac{1}{X^2 (1 + \frac{2\cdot dx}{X} + \frac{dx^2}{X^2})}\)
\(\displaystyle \dfrac{1}{X^2 (1 + \frac{dx}{X})^2}\)
\(\displaystyle X^{-2}(1 + \frac{dx}{X})^{-2}\)