# using the total derivative to find a minimum value

I have a function T(x,y) where x and y are functions of t. I am given the two first order partial derivatives (dt/dx and dt/dy - the d is meant to be the partial symbol) and I am asked to use the total derivative to find a minimum value of t.

I have found the total derivative using the chain rule but I am unsure how this helps me find a minimum for t?

edit: all information given: dt/dx=6x-3y dt/dy=6y-3x x=cos(t) y=sin(t) *t between 0 and pi/2

• Could you write in all information you have? – Mark Fantini Dec 9 '13 at 19:46
• dt/dx=6x-3y dt/dy=6y-3x x=cost y=sint 0<t<pi/2 – maths Dec 9 '13 at 19:47
• That would be best done in the question itself. =) Also, the $t$ you are constantly referring is the function $T(x,y)$ or the parameter $t$? – Mark Fantini Dec 9 '13 at 19:47
• it is the parameter – maths Dec 9 '13 at 19:49
• I think what you are writing as $$\frac{dt}{dx}$$ is actually $$\frac{\partial T}{\partial x}.$$ – Mark Fantini Dec 9 '13 at 19:51

You have $T(x,y)$ where $x$ and $y$ are functions of a parameter $t$. In other words, we can abuse notation and write $T(x,y) = T(x(t), y(t)) = T(t)$. Using the chain rule, we get
$$\frac{dT}{dt} = \frac{\partial T}{\partial x} \frac{dx}{dt} + \frac{\partial T}{\partial y} \frac{dy}{dt}.$$