Applications of Derivatives

The temperature $T$ (in °C) of a chemical mixture $x$ minutes after a catalyst is added is modeled by $T(x) = \frac{x^2 \sqrt{x+1}}{(x-2)^2}$. Determine the slope of the tangent line to the temperature-time graph when $x = 3$ minutes.

A curve is defined by the equation $y = \frac{\ln(x^2+1)}{e^{2x}}$. Find the slope of the tangent line to the curve at the point where $x = 1$.

The concentration of a drug in the bloodstream $t$ hours after injection is given by $C(t) = \frac{5t}{t^2 + 1}$ mg/L. Find the slope of the tangent line to the graph of $C(t)$ at $t = 2$ hours.