How to Find dz/dt in a Derivative with 3 Variables?

[Runaway]

1/z = 1/x + 1/y
If x is increasing at 4 units/s and y is increasing at 6 units/s, how fast is z increasing when x=20 and y=30?

My question is how do I find an equation for dz/dt? Does it equal dx/dt + dy/dt?

 

[rock.freak667]

You need to apply something like this


d/dt(f(x)) = (dx/dt)*d/dx(f(x)) = (dx/dt)f'(x)

Similar for f(y).

 

[Mark44]

Or just differentiate implicitly with respect to t. On the left side you would have
d/dt(1/z) = d/dz(1/z) * dz/dt = -1/z² * dz/dt.

dz/dt won't be equal to dx/dt + dy/dt. The relationship between x and y is nonlinear, so you won't get dx/dt + dy/dt when you differentiate the right side.

 

[MysticDude]

Or just differentiate implicitly with respect to t. On the left side you would have
d/dt(1/z) = d/dz(1/z) * dz/dt = -1/z² * dz/dt.

dz/dt won't be equal to dx/dt + dy/dt. The relationship between x and y is nonlinear, so you won't get dx/dt + dy/dt when you differentiate the right side.

To continue on with what Mark44 said: $$\frac{dz}{dt}\frac{-1}{z^2} = \frac{dx}{dt}\frac{-1}{x^2} + \frac{dy}{dt}\frac{-1}{y^2}$$ You have 2 unknowns, one is dz/dt which is what you are looking for and the other is z itself at that time. How would you find z? Hint: It's in your question!