Question

Find the derivative of the given function.
$f(x) = 10 \sqrt[5]{ {x}^{3} } - \sqrt{ {x}^{7} } + 6 \sqrt[3]{ {x}^{8} } - 3$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\rm \: f(x) = 10 \sqrt[5]{ {x}^{3} } - \sqrt{ {x}^{7} } + 6 \sqrt[3]{ {x}^{8} } - 3$

can be rewritten as

$\rm \: f(x) = 10 {\bigg( {x}^{3} \bigg) }^{\dfrac{1}{5} } - {\bigg( {x}^{7} \bigg) }^{\dfrac{1}{2} } + 6{\bigg( {x}^{8} \bigg) }^{\dfrac{1}{3} } - 3$

can be further rewritten as

$\rm \: f(x) = 10 {\bigg(x\bigg) }^{\dfrac{3}{5} } - {\bigg(x\bigg) }^{\dfrac{7}{2} } + 6{\bigg(x\bigg) }^{\dfrac{8}{3} } - 3$

On differentiating both sides w. r. t. x, we get

$\rm \:\dfrac{d}{dx}f(x) =\dfrac{d}{dx}\bigg(10 {\bigg(x\bigg) }^{\dfrac{3}{5} } - {\bigg(x\bigg) }^{\dfrac{7}{2} } + 6{\bigg(x\bigg) }^{\dfrac{8}{3} } - 3\bigg)$

$\rm \: f'(x) = 10\dfrac{d}{dx} {\bigg(x\bigg) }^{\dfrac{3}{5} } - \dfrac{d}{dx}{\bigg(x\bigg) }^{\dfrac{7}{2} } + 6\dfrac{d}{dx}{\bigg(x\bigg) }^{\dfrac{8}{3} } - \dfrac{d}{dx}3$

We know,

$\color{green}\boxed{ \rm{ \:\dfrac{d}{dx} {x}^{n} \: = \: {nx}^{n - 1} \: \: }}$

So, using this result, we get

$\rm \: f'(x) = 10 \times \dfrac{3}{5} {\bigg(x\bigg) }^{\dfrac{3}{5} - 1} - \dfrac{7}{2}{\bigg(x\bigg) }^{\dfrac{7}{2} - 1} + 6 \times \dfrac{8}{3}{\bigg(x\bigg) }^{\dfrac{8}{3} - 1} - 0$

$\rm \: f'(x) = 6 {\bigg(x\bigg) }^{ - \dfrac{2}{5}} - \dfrac{7}{2}{\bigg(x\bigg) }^{\dfrac{5}{2}} + 16{\bigg(x\bigg) }^{\dfrac{5}{3} }$

Hence,

$\color{green}\rm\implies \:\boxed{ \rm{f'(x) = 6 {\bigg(x\bigg) }^{ - \dfrac{2}{5}} - \dfrac{7}{2}{\bigg(x\bigg) }^{\dfrac{5}{2}} + 16{\bigg(x\bigg) }^{\dfrac{5}{3} }}}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {x}^{n}\\ \\ \sf {nx}^{n - 1} & \sf {e}^{x} \end{array}} \\ \end{gathered}$

Answer 2

refer the given attachment