Question

Differentiate the function %7Bx%7D%20" id="TexFormula1" title="7 {x}^{3} + x \div 6 \sqrt{x} " alt="7 {x}^{3} + x \div 6 \sqrt{x} " align="absmiddle" class="latex-formula">

Answer 1

There are two likely interpretations for the given function:

$f(x)=7x^3+\dfrac x{6\sqrt x}$

in which case you can write

$\dfrac x{6\sqrt x}=\dfrac x{6x^{1/2}}=\dfrac16x^{1/2}$

and the derivative is

$f'(x)=21x^2+\dfrac1{12}x^{-1/2}=21x^2+\dfrac1{12\sqrt x}$

Or, perhaps it is

$f(x)=\dfrac{7x^3+x}{6\sqrt x}=\dfrac76x^{5/2}+\dfrac16x^{1/2}$

$\implies f'(x)=\dfrac{35}{12}x^{3/2}+\dfrac1{12\sqrt x}$