Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\log _{ a }{ x } \) then \(\frac { dy }{ dx } \) is ______

(a)

\(\frac { 1 }{ x } \)

(b)

\(\frac { 1 }{ x\log _{ e }{ a } } \)

(c)

\(\log { _{ e }^{ a } } \)

(d)

\(\frac { 1 }{ \log _{ a }{ x } } \)

Choose the correct or the most suitable answer from the given four alternatives.
The derivative of \(\cos ^{ -1 }{ (2{ x }^{ 2 } } -1)\) with respect to \(\cos ^{ -1 }{ x } \) is _____

(a)

2

(b)

\(\frac { 1 }{ 2\sqrt { 1-{ x }^{ 2 } } } \)

(c)

\(\frac { 2 }{ x } \)

(d)

\(1-{ x }^{ 2 }\)

Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\log \left(\frac{1-x^2}{1+x^2}\right)\) then \(\frac{dy}{dx}\) is ______

(a)

\(\frac { 4{ x }^{ 3 } }{ 1-{ x }^{ 4 } } \)

(b)

\(-\frac { 4x }{ 1-{ x }^{ 4 } } \)

(c)

\(\frac { 1 }{ 4-{ x }^{ 4 } } \)

(d)

\(\frac { -4{ x }^{ 3 } }{ 1-{ x }^{ 4 } } \)

Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\sin ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\) then \(\frac{dy}{dx}\) is ______

(a)

\(\frac { -2 }{ 1+{ x }^{ 2 } } \)

(b)

\(\frac { 2 }{ 1+{ x }^{ 2 } } \)

(c)

\(\frac { 1 }{ 2-{ x }^{ 2 } } \)

(d)

\(\frac { 2 }{ 2-{ x }^{ 2 } } \)

Choose the correct or the most suitable answer from the given four alternatives.
If, \(y=a+b{ x }^{ 2 }\) where a, b are arbitrary constants, then ____

(a)

\(\frac { d^{ 2 }y }{ d{ x }^{ 2 } } =2xy\)

(b)

\(x\frac { d^{ 2 }y }{ d{ x }^{ 2 } } ={ y }_{ 1 }\)

(c)

\(x\frac { d^{ 2 }y }{ d{ x }^{ 2 } } -\frac { dy }{ dx } +y=0\)

(d)

\(x\frac { d^{ 2 }y }{ d{ x }^{ 2 } } =2xy\)