Differential Calculus - Differentiability and Methods of Differentiation Three Marks Questions

11th Standard

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
    10 x 3 = 30
  1. 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.

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

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

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

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

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

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

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

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

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

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