linguistic:translation, quality:correction
Calculus

In real calculus, the derivative of 1/ x = x is given by the power rule with the power −1:

The power rule for integrals (Cavalieri's quadrature formula) cannot be used to compute the integral of 1/ x, because doing so would result in division by 0: ∫ d x x = x 0 0 + C {\displaystyle \int {\frac {dx}{x}}={\frac {x^{0}}{0}}+C} Instead the integral is given by: ∫ 1 a d x x = ln a, {\displaystyle \int _{1}^{a}{\frac {dx}{x}}=\ln a,} ∫ d x x = ln x + C. {\displaystyle \int {\frac {dx}{x}}=\ln x+C.} where ln is the natural logarithm. To show this, note that d d y e y = e y {\textstyle {\frac {d}{dy}}e^{y}=e^{y}}, so if x = e y {\displaystyle x=e^{y}} and y = ln x {\displaystyle y=\ln x}, we have: d x d y = x ⇒ d x x = d y ⇒ ∫ d x x = ∫ d y = y + C = ln x + C. {\displaystyle {\begin{aligned}&{\frac {dx}{dy}}=x\quad \Rightarrow \quad {\frac {dx}{x}}=dy\\&\quad \Rightarrow \quad \int {\frac {dx}{x}}=\int dy=y+C=\ln x+C.\end{aligned}}}