Which of the following is/are correct :

The correct option is C $S_3:$ Derivative of $5^x+{\log }_5{x}^2$ w.r.t. $x$ is $5^x{\log }_{e}5+\frac{2}{x{\log }_{e}5}$
$S_1:$ Derivative of ${\sec }^{-1}x$ w.r.t. $x$ is $\frac{1}{|x|\sqrt{x^2-1}},|x|>1$
Hence, $S_1$is true.

$S_2:$ Derivative of $\frac{1}{x^{207}}$ w.r.t. $x$ is $\frac{-207}{x^{208}}$
Hence, $S_2$ is true.

$S_3:$ Derivative of $5^x+{\log }_5{x}^2$ w.r.t. $x$ is $5^x{\log }_{e}5+\frac{2}{x{\log }_{e}5}$
Hence, $S_3$ is true.

$S_4:$ Derivative of $x^{99}\sin x$ by product rule is $99x^{98}\sin x+x^{99}\cos x$
Hence, $S_4$ is false.