Derivatives

(cod: P-52-32-5) Consider the following statements:

(1) If \(f(x) = x^r\), for \(x > 0\), where \(r\) is any real number, then \(f'(x) = r x^{r-1}\).

(2) \(\dfrac{d}{dx}\ln(|x|) = \dfrac{1}{x}.\)

(3) \(\dfrac{d}{dx} \sinh\, x = \cosh{x}.\)

(4) \(\dfrac{d}{dx} \cosh{x} = - \sinh\, {x}.\)

(5) \(\dfrac{d}{dx} \tanh{x} = \operatorname{sech}^2\, {x}.\)

(6) \(\dfrac{d}{dx} \operatorname{sech}\, {x} = \operatorname{sech}\, {x} \tanh{x}.\)

Regarding these statements, we have that:

The only false statements are (4) and (6).

The only false statements are (2) and (3).

The only false statements are (1) and (5).

The only false statement is (2).

The only false statements are (2) and (4).