Question

Differentiate the following functions with respect to x :

${(\sqrt{x}+\frac{1}{\sqrt{x}})}^3$

Answer 1

We have,

$\frac{d}{dx}{(\sqrt{x}+\frac{1}{\sqrt{x}})}^3$

$=\frac{d}{dx}(x^{\frac{3}{2}}+3x.\frac{1}{\sqrt{x}}+3\sqrt{x}.\frac{1}{x}+\frac{1}{x^{\frac{3}{2}}})\left[\right(a+b{)}^3=a^2+3a^{2}b+3a{b}^2+b^3]$

$=\frac{d}{dx}(x^{\frac{3}{2}}+3x^{\frac{1}{2}}+3x^{\frac{-1}{2}}+x^{\frac{-3}{2}})=\frac{3}{2}{x}^{\frac{1}{2}}+\frac{3}{2}{x}^{\frac{-1}{2}}+3.\left(\frac{-1}{2}\right)x^{\frac{-1}{2}}-\frac{3}{2}{x}^{\frac{-5}{2}}$

$=\frac{3}{2}{x}^{\frac{1}{2}}-\frac{3}{2}{x}^{\frac{-5}{2}}+3x^{\frac{-1}{2}}-\frac{3}{2}{x}^{\frac{-3}{2}}$