[tex]DIFFERENTIATION \\ \\ \\ Let \: u \: = \: {x}^{6} \: \: \: and \: v = \frac{1}{ \sqrt{x} } \\ \\ Then \: , \: \frac{du}{dx} \: = \: 6 {x}^{5} \\ \\ and \: \: \: \: \: \: \frac{dv}{dx} \: \: = \: \frac{ - 1}{2x \sqrt{x} } \\ \\ \\ Therefore \: , \\ \\ \\ \frac{du}{dv} \: = \: \frac{ \frac{du}{dx} }{ \frac{dv}{dx} } \: = \: \frac{6 {x}^{5} }{ \frac{ - 1}{2x \sqrt{x} } } \\ \\ \frac{du}{dv} \: = \: \frac{ \frac{du}{dx} }{ \frac{dv}{dx} } \: = - 12 {x}^{ \frac{13}{2} } \\ \\ \\ \frac{du}{dv} \: = - 12 {x}^{ \frac{13}{2} } \: \: \: \: \: \: \: \: Ans.[/tex]
