Show that the function \(f\left( x \right) =\begin{cases} x-1,\quad x<2 \\ 2x-3,\quad x\ge 2 \end{cases}\)is not differentiable at x = 2.

Show  that\(f\left( x \right) ={ x }^{ 2 }\) is differentiable at x = 1 and find \(f^{ ' }\left( 1 \right) \)

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

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

If xy = 4, Prove that \(x\left( \frac { dy }{ dx } +{ y }^{ 2 } \right) =3y.\)

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

If \({ x }^{ 2 }+2xy+{ y }^{ 3 }=42,\) find \(\frac { dy }{ dx } \)

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

Differentiate \(\log { (1+{ x }^{ 2 } } )\) with respect to \(\tan ^{ -1 }{ x } \)

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