How do you use the definition of derivative to compute the derivative of the function #f(x) = (x) / (3x + 1)#?

To compute the derivative of the function ( f(x) = \frac{x}{3x + 1} ), you can use the definition of the derivative:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Substitute ( f(x) = \frac{x}{3x + 1} ) into the formula:

[ f'(x) = \lim_{h \to 0} \frac{\frac{x + h}{3(x + h) + 1} - \frac{x}{3x + 1}}{h} ]

Simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{\frac{x + h}{3x + 3h + 1} - \frac{x(3x + 1)}{(3x + 1)(3x + 1)}}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{(x + h)(3x + 1) - x(3x + 1)}{h(3x + 3h + 1)(3x + 1)} ]

[ f'(x) = \lim_{h \to 0} \frac{3x^2 + xh + 3x + h - 3x^2 - x}{h(3x + 3h + 1)(3x + 1)} ]

[ f'(x) = \lim_{h \to 0} \frac{xh + h}{h(3x + 3h + 1)(3x + 1)} ]

[ f'(x) = \lim_{h \to 0} \frac{h(x + 1)}{h(3x + 3h + 1)(3x + 1)} ]

[ f'(x) = \lim_{h \to 0} \frac{x + 1}{(3x + 1)(3x + 1)} ]

Evaluate the limit as ( h \to 0 ):

[ f'(x) = \frac{x + 1}{(3x + 1)^2} ]