Differentiate \(f\left( x \right) ={ e }^{ 2x }\)from first principles.

If \(y=\sqrt { x+1 } +\sqrt { x-1 } \) prove that\(\sqrt { { x }^{ 2 }+1 } \frac { dy }{ dx } =\frac { 1 }{ 2 } y.\)

Differentiate \((\sec ^{-1}\left(\frac{1}{2 x^{2}-1}\right), \quad 0)\)

If x = \(a\sec ^{ 3 }{ \theta }\) and \(y=a\tan ^{ 3 }{ \theta }\) find \(\frac { dy }{ dx }\) at \(\theta =\frac { \pi }{ 3 }\)

If f(x) = 2x² + 3x - 5, then prove that f' (0) + 3 f' (-1) = 0