All powers of x are constant.

Proposition. Let n be a nonnegative integer. The function xn is constant.

Proof. Observe that (x0)′ = 0. Assume that the derivative of xn is zero for n = 0, 1, 2, …, k. Then

(xk+1)′ = (x · xk)′ = x′ · xk + x · (xk)′

is also zero since x′ = (x1)′ = (xk)′ = 0. ♡

Contributed by Alex Kuperman of the Israel Institute of Technology (Technion) in Haifa.

Differentiating the square function

At x = c, the function y = (x − c)2 = x2 − 2cx + c2 has a minimum, so that 0 = Dy = D(x2) − 2cD(x) = D(x2) − 2c. But c is arbitrary and c = x. Hence D(x2) = 2c = 2x.

Contributed by A.W. Walker of Toronto, ON.

equals 2

Let x be positive. Differentiating the equation x3 = x2 + x2 + … + x2 (to x terms) yields 3x2 = 2x + 2x + … + 2x = x(2x) = 2x2, whence 3 = 2. ♢

An alternative proof of the same fact goes like this. Let x be constant with the value 1. Then x = x2 = x3. Now set y = x. Then y = x2 and y = x3. Therefore dy/dx = 2x and dy/dx = 3x2. Therefore dy/dx is both constant with value 2 and constant with value 3. ♢