#### 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 ${ x }^{ 2 }+2xy+{ y }^{ 3 }=42,$ find $\frac { dy }{ dx }$

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

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