Multivariable Differentials

I did single variable differentials a while ago here: http://www.mathwizurd.com/calc/2015/11/18/differentials

Multivariable Differentials

Recall that in single variable, if $u = f(x)$, then $du = f'(x)dx$. We can extend this to multiple variables.
If $w = f(x,y)$, $$dw = f_x(x,y)dx + f_y(x,y)dy = \frac{\partial{}w}{\partial{}x}dx + \frac{\partial{}w}{\partial{}y}dy$$ Similar to single variable differentials, this also finds a tangent line to the 3D curve expressed by the function and approximates $w + \Delta{}w$.

Example

Estimate $\sqrt{225.08}(2.003)^3$ Immediately, we have to figure out x, y, dx, dy, and the function. $$f(x,y) = \sqrt{x}(y)^3$$ $$x = 225, dx = 0.08$$ $$y = 2, dy = 0.003$$ $$dw = f_x(x,y)(0.08) + f_y(x,y)(0.003)$$ $$dw = \dfrac{y^3}{2\sqrt{x}}(0.08) + 3y^2\sqrt{x}(0.003)$$ $$dw = \dfrac{8}{30}(0.08) + 180(0.003) = 0.56133$$ $$w + dw \approx f(x + dx, y + dy, z + dz)$$ $$w + dw = \boxed{120.561333}$$ Real answer: $120.56223$, so very close