What is the constant of integration for each of the above integrals? Explain.

What do you notice about the above results? Explain.

Total Differential

Definition: Total Differential

Recall from Calculus III:

Suppose some function $F$ is dependent on two independent variables, $x$ and $y$, more specifically, $F(x,y)=C$, where $C$ is a constant. If $F$ is differentiable, then the total differential is: $dF={\displaystyle \frac{\partial F}{\partial x}}dx+{\displaystyle \frac{\partial F}{\partial y}}dy$

Investigation 03

Find the total differential of ${x}^{2}y+x\mathrm{sin}(y)-x+{e}^{y}=7$.

Let ${\displaystyle \frac{\partial F}{\partial x}}=M(x,y)$ and ${\displaystyle \frac{\partial F}{\partial y}}=N(x,y)$.