List of top Higher Mathematics Questions on Differentiation

If \( y = \sqrt{e^x} \), \( x > 0 \), then \( \frac{dy}{dx} = \underline{\hspace{2cm}} \)

Show that the function given by \( f(x) = e^{3x} \) is increasing on \( R \).

Differentiate \( \sin(x^2) \) with respect to \( x^2 \).
Differentiate \( X^X \) with respect to \( X \).
If \( y = \sin^{-1} \left( \frac{2x}{1 + x^2} \right) \), then find \( \frac{dy}{dx} \).
Show that the function given by \( f(x) = 12x - 3 \) is increasing on \( R \).