Answer
$$\eqalign{
& \left( a \right).{\text{ }}m = - \frac{3}{4} \cr
& \left( b \right).{\text{ }}y = - \frac{3}{4}x + \frac{1}{2} \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{1}{{2\sqrt {3x + 1} }};{\text{ }}P\left( {0,\frac{1}{2}} \right) \cr
& \left( a \right){\text{ Calculate }}f'\left( x \right){\text{ using the definition of the derivative}} \cr
& f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{\frac{1}{{2\sqrt {3\left( {x + h} \right) + 1} }} - \frac{1}{{2\sqrt {3x + 1} }}}}{h} \cr
& f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{\frac{1}{{2\sqrt {3x + 3h + 1} }} - \frac{1}{{2\sqrt {3x + 1} }}}}{h} \cr
& f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{2\sqrt {3x + 1} - 2\sqrt {3x + 3h + 1} }}{{\left( {2\sqrt {3x + 3h + 1} } \right)\left( {2\sqrt {3x + 1} } \right)h}} \cr
& {\text{Rationalizing the numerator}} \cr
& f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{2\sqrt {3x + 1} - 2\sqrt {3x + 3h + 1} }}{{\left( {2\sqrt {3x + 3h + 1} } \right)\left( {2\sqrt {3x + 1} } \right)h}} \times \frac{{2\sqrt {3x + 1} + 2\sqrt {3x + 3h + 1} }}{{2\sqrt {3x + 1} + 2\sqrt {3x + 3h + 1} }} \cr
& f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{{{\left( {2\sqrt {3x + 1} } \right)}^2} - {{\left( {2\sqrt {3x + 3h + 1} } \right)}^2}}}{{\left( {2\sqrt {3x + 3h + 1} } \right)\left( {2\sqrt {3x + 1} } \right)h\left( {2\sqrt {3x + 1} + 2\sqrt {3x + 3h + 1} } \right)}} \cr
& f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{4\left( {3x + 1} \right) - 4\left( {3x + 3h + 1} \right)}}{{\left( {2\sqrt {3x + 3h + 1} } \right)\left( {2\sqrt {3x + 1} } \right)h\left( {2\sqrt {3x + 1} + 2\sqrt {3x + 3h + 1} } \right)}} \cr
& f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{12x + 4 - 12x - 12h - 4}}{{\left( {2\sqrt {3x + 3h + 1} } \right)\left( {2\sqrt {3x + 1} } \right)h\left( {2\sqrt {3x + 1} + 2\sqrt {3x + 3h + 1} } \right)}} \cr
& f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{ - 12h}}{{\left( {2\sqrt {3x + 3h + 1} } \right)\left( {2\sqrt {3x + 1} } \right)h\left( {2\sqrt {3x + 1} + 2\sqrt {3x + 3h + 1} } \right)}} \cr
& f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{ - 12}}{{\left( {2\sqrt {3x + 3h + 1} } \right)\left( {2\sqrt {3x + 1} } \right)\left( {2\sqrt {3x + 1} + 2\sqrt {3x + 3h + 1} } \right)}} \cr
& {\text{ }}h \to 0 \cr
& f'\left( x \right) = \frac{{ - 12}}{{\left( {2\sqrt {3x + 3\left( 0 \right) + 1} } \right)\left( {2\sqrt {3x + 1} } \right)\left( {2\sqrt {3x + 1} + 2\sqrt {3x + 3\left( 0 \right) + 1} } \right)}} \cr
& f'\left( x \right) = \frac{{ - 12}}{{\left( {2\sqrt {3x + 1} } \right)\left( {2\sqrt {3x + 1} } \right)\left( {2\sqrt {3x + 1} + 2\sqrt {3x + 1} } \right)}} \cr
& f'\left( x \right) = \frac{{ - 12}}{{4\left( {3x + 1} \right)\left( {4\sqrt {3x + 1} } \right)}} \cr
& f'\left( x \right) = - \frac{3}{{4{{\left( {3x + 1} \right)}^{3/2}}}} \cr
& {\text{Calculate the slope at }}P\left( {0,\frac{1}{2}} \right) \cr
& m = f'\left( 0 \right) = - \frac{3}{{4{{\left( {3\left( 0 \right) + 1} \right)}^{3/2}}}} = - \frac{3}{4} \cr
& \cr
& \left( b \right){\text{ Find an equation of the tangent line at }}P\left( {0,\frac{1}{2}} \right) \cr
& y - \frac{1}{2} = - \frac{3}{4}\left( {x - 0} \right) \cr
& {\text{ }}y = - \frac{3}{4}x + \frac{1}{2} \cr
& {\text{ }}y = - \frac{3}{4}x + \frac{1}{2} \cr} $$
