Answer
$y'==\dfrac{t^{4}-8t^{3}+6t^{2}+9}{(t^{2}-4t+3)^{2}}$
Work Step by Step
$y=\dfrac{t^{3}+3t}{t^{2}-4t+3}$
Differentiate using the quotient rule:
$y'=\dfrac{(t^{2}-4t+3)(t^{3}+3t)'-(t^{3}+3t)(t^{2}-4t+3)'}{(t^{2}-4t+3)^{2}}=...$
$...=\dfrac{(t^{2}-4t+3)(3t^{2}+3)-(t^{3}+3t)(2t-4)}{(t^{2}-4t+3)^{2}}=...$
Evaluate the products and simplify:
$...=\dfrac{3t^{4}+3t^{2}-12t^{3}-12t+9t^{2}+9-2t^{4}-6t^{2}+4t^{3}+12t}{(t^{2}-4t+3)^{2}}=...$
$...=\dfrac{t^{4}-8t^{3}+6t^{2}+9}{(t^{2}-4t+3)^{2}}$
