Problem

Which of the following statement is false?

(a) $\dfrac{d}{dx}\ln(2x)=\dfrac{d}{dx}\ln(x)$ for $x>0$.

(b) $\dfrac{d}{dx}\csc(x)=\cot(x)\csc(x)$ for $x\in(0,\pi)$.

(c) $\dfrac{d}{dx}\dfrac{\sin(x)}{\cos(x)}=1+\tan^2(x)$ for $x\in(-\pi/2,\pi/2)$.

(d) $\dfrac{d}{dx}2^x=\ln(2)\cdot 2^x$ for $x\in(-\infty,\infty)$.

(e) $\dfrac{d}{dx}(x^3+2x+1)=3x^2+2$ for $x\in(-\infty,\infty)$.