[Math] Using the weekly sales function, find the rate at which weekly sales are changing

Question

Using the weekly sales function $s=1980+45x+0.65x^2$ with $x$ representing weekly advertising costs, find the rate at which the weekly sales are changing when the weekly advertising costs are 8400 dollars and these costs are increasing at a rate of 1250 dollars per week

Answer 1

Let us define the rate at which weekly sales are changing to be $\frac{dS}{dT}$

Let us define the rate at which weekly advertising costs are changing to be $\frac{dx}{dT}$

Let us define the change in weekly sales with respect to change in advertising costs to be $\frac{dS}{dx}$

$S=1980+45x+.65x^2$

$\frac{dS}{dT}$ = $\frac{dS}{dx}$*$\frac{dx}{dT}$

$\frac{dS}{dT}$ = $(45+1.3x)$*$\frac{dx}{dT}$

Now evaluate $\frac{dS}{dT}$

at x = $8400, $$\frac{dx}{dT}$ = 1250