Questions

1. Question 1

   Differentiate w.r.t ‘x’.

   (i)

   $x^4 + 2x^3 + x^2$

   (ii)

   $x^{-3} + 2x^{-\frac{3}{2}} + 3$

   (iii)

   $\dfrac{2x - 3}{2x + 1}$

   (iv)

   $\dfrac{(1 + \sqrt{x})(x - x)}{\sqrt{x}}$

   (v)

   $\left( \sqrt{x} - \dfrac{1}{\sqrt{x}} \right)^2$

   (vi)

   $(x - 5)(3 - x)$

   (vii)

   $\dfrac{(x^2 + 1)^2}{x^2 - 1}$

   (viii)

   $\dfrac{x^2 + 1}{x^2 - 3}$

   (ix)

   $\dfrac{2x - 1}{\sqrt{x^2 + 1}}$

   (x)

   $\sqrt{\dfrac{a - x}{a + x}}$

   (xi)

   $\dfrac{\sqrt{x^2 + 1}}{\sqrt{x^2 - 1}}$

   SolutionTheory
2. Question 2

   Find $\dfrac{dy}{dx}$ if $y = \dfrac{(\sqrt{x} + 1)(x^{\frac{3}{2}} - 1)}{x^{\frac{1}{2}} - 1}$, $(x \ne 1)$.

   SolutionTheory
3. Question 3

   Differentiate $\dfrac{(\sqrt{x} + 1)(x^{\frac{3}{2}} - 1)}{x^{\frac{3}{2}} - x^{\frac{1}{2}}}$ with respect to $x$.

   SolutionTheory
4. Question 4

   If $y = \sqrt{x} - \dfrac{1}{\sqrt{x}}$, show that $2x \dfrac{dy}{dx} + y = 2\sqrt{x}$.

   SolutionTheory
5. Question 5

   If $y = x^4 + 2x^2 + 2$, prove that $\dfrac{dy}{dx} = 4x\sqrt{y - 1}$.

   SolutionTheory