Find the equation of the tangent line to the graph of the function at z=3
Find the derivative of the function using the quotient rule:
$\displaystyle f(x) = \dfrac{6z^2}{5z^2 + 2z}$
$\displaystyle f'(z) = \dfrac{(5z^2 + 2z)(12z) - (6z^2)(10z + 2)}{(5z^2 + 2z)^2}$
Put z = 3.
$\displaystyle f'(3) = \dfrac{(5(3)^2 + 2(3))(12(3)) - (6(3)^2)(10(3) + 2)}{(5(3)^2 + 2(3))^2}$
Simplify, then find the value of f(z) when z = 3.
Then use:
$\displaystyle y = mx+ c$
m is f'(3), and c is a constant.
To find c, use y = f(3) and x = z
You'll get your equation of the line now. You can replace y and x by f(z) and z respectively.