# [Math] Using the Cost and Revenue Function, find the rate at which profit is changing.

calculusfunctions

Using the cost function $c=190,000+0.75x$ and the revenue function $r=180x-0.125x^2$, find the rate at which the profit is changing when the production is $200$ units and the rate of change of production is $150$ units per week?

Let be $x=x(t)$ the production at time $t$, i.e. the units produced at time $t$, and $\frac{\operatorname{d}x}{\operatorname{d}t}=x'(t)$ the rate of change of production per week. Let be $c(x)=c_0+c_1x$ the cost function and $r(x)=r_1x-r_2x^2$ the revenue function with $c_0=190,000$, $c_1=0.75$, $r_1=180$, $r_2=0.125$.
Let be $\pi(x)=r(x)-c(x)$ the profit function, $x_0=200$ and $\frac{\operatorname{d}x}{\operatorname{d}t}=x'(t)=150$ the rate of change of production per week. So, you have to find the rate of change of $\pi(x(t))$ as the time varies, that is $$\frac{\operatorname{d}\pi(x(t))}{\operatorname{d}t}=\pi'(t)=\left[r'(x)-c'(x)\right]x'(t)=\left[(r_1-c_1)-2r_2x(t)\right]x'(t)$$ and evaluate it at time $t=t_0$ for which $x(t_0)=x_0$ and $x'(t)=150$: $$\left.\frac{\operatorname{d}\pi(x(t))}{\operatorname{d}t}\right|_{t=t_0}=\pi'(t_0)=19,387.5$$