# Thread: Can anyone solve this Calculus 3 rate of change problem?

1. ## Can anyone solve this Calculus 3 rate of change problem?

The temperature at a point (x,y) on a flat metal plate is given by T(x,y)=20/(1+x^4+y^2) where T is measured in degrees Celsius and x,y in meters. Find the rate of change of temperature with respect to the distance at the point (1,2) in the direction of the unit vector

u=<4/5,3/5>

2. ## Re: Can anyone solve this Calculus 3 rate of change problem?

dT(1,2)/dx = -20/9
dT(1,2)/dy = -20/9

dT={-20/9, -20/9} as vector
u={4/5, 3/5} unit vector

dot product dT.u = -28/9

3. ## Re: Can anyone solve this Calculus 3 rate of change problem?

Hey, MaxJasper, what do you use to create those pretty pictures?

To allstar2 (mostly just re-hashing what MaxJasper wrote) - the thing you're looking for is called a directional derivative, and it is equal to $\displaystyle \nabla{T}\cdot{u}$, u being a unit vector in the direction you want to take the derivative. If you write it out, you get $\displaystyle \frac{4}{5}\frac{dT}{dx}+\frac{3}{5}\frac{dT}{dy}$, which is what you'd write if x and y were functions of some other variable t and you applied the chain rule.

- Hollywood