For $\small{f(x) = \frac{15}{ x}}$, find $\large{\frac{dy}{dx}}$ using the definition, $$\small{\frac{dy}{dx} = \displaystyle\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \displaystyle\lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}.}$$

(a) First, evaluate $\small{\Delta y = f(x + \Delta x) - f(x)}$ and express it in the form $\small{\Delta y = L(x,\Delta x) \cdot \Delta x}$. Use $\small{dx}$ to represent $\small{\Delta x}$.

$\small{\Delta y}$ = $\small{\Delta x}$

(b) Using the $\small{L(x,\Delta x)\; \Delta x} \;$ above, find the simplified derivative $\large{\frac{dy}{dx}} \small{= \displaystyle\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}}$.

$\large{\frac{dy}{dx}} \small{= \displaystyle\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} =}$