Prove that the identity function on real numbers given by: f(x) = x is continuous at every real number.

3

(a)

Discuss the continuity of the function f defined by:
\(f(x)={ x }^{ 3 }+{ x }^{ 2 }-1\)

3

(a)

Find the derivative of tan(2x+3).

3

(a)

Differentiate \(sin(cos({ x }^{ 2 }))\quad with\quad respect\quad to\quad x.\)

3

(a)

\(Find\ \frac { dy }{ dx } \ if\ x-y=\pi \)

3

(a)

Find the values of 'p' and 'q', for which:
\(\begin{cases} \frac { 1-{ sin }^{ 3 }x }{ 3{ cos }^{ 2 }x } ,\quad if\quad x<\frac { \pi }{ 2 } \\ p,\quad \quad \quad \quad \quad \quad \quad if\quad x=\frac { \pi }{ 2 } \\ \frac { q(1-sinx) }{ { \left( \pi -2x \right) }^{ 2 } } ,\quad if\quad x>\frac { \pi }{ 2 } \end{cases}\)
is continuous at x = 2.

3

(a)

If \(y={ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } } \right) ,{ x }^{ 2 }\le 1\) then find \(\frac { dy }{ dx } \)

3

(a)

Find \(\frac { dy }{ dx } \) if \(y={ sin }^{ -1 }\left[ \frac { 6x-4\sqrt { 1-4{ x }^{ 2 } } }{ 5 } \right] \) 

3

(a)

If \({ x }^{ y }={ e }^{ x-y }\)show that \(\frac { dy }{ dx } =\frac { log\quad x }{ { \left( log(xe) \right) }^{ 2 } } \)

3

(a)

Differentiate \(log({ x }^{ x }+{ cosec }^{ 2 }x)\) w.r.t. x.

3

(a)