CLEP Calculus - How to find differentiable of rate

Find the rate of change of $y=4x^{3}-12x^{2}+6x+7$ at .

The find the change of volume of a spherical balloon that is growing from to

For the relation $\small x^2y+2x-y^2=x$ , compute $\small y'$ using implicit differentiation.

Let $y = x^{xlnx-e}.$ Use logarithmic differentiation to find .

In chemistry, rate of reaction is related directly to rate constant .

$\frac{dC}{dt}=k$ , where is the initial concentration

Give the concentration of a mixture with rate constant $k=\frac{3}{sec}$ and initial concentration , seconds after the reaction began.

We can interperet a derrivative as $\frac{dy}{dx}= \lim_{\Delta \rightarrow 0} \frac{\Delta y}{\Delta x}$ (i.e. the slope of the secant line cutting the function as the change in x and y approaches zero) but these so-called "differentials" ( $\Delta x$ and $\Delta y$ ) can be a good tool to use for aproximations. If we suppose that $\frac{dy}{dx}=f'(x)$ , or equivalently . If we suppose a change in x (have a concrete value for $\Delta x$ ) we can find the change in with the afore mentioned relation.