If x\(\sqrt { 1+y } +y\sqrt { 1+x } =0\) and x ≠ y, then prove that \(\frac { dy }{ dx } =\frac { -1 }{ (x+1)^{ 2 } } \).

Differentiate the following with respect to x \(\sqrt { \frac { (x-1)(x-2) }{ (x-3)({ x }^{ 2 }+x+1) } } \)

Show that \( f(x)= \begin{cases}5 x-4, & \text { if } 0< x \leq 1 \\ 4 x^3-3 x, & \text { if } 1< x< 2\end{cases}\) is continuous at x = 1

Show that the function f(x) = |x| is not differentiable at x = 0.

Differentiate: \(\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\)