1. **Problem Statement:** Understand the difference between partial derivatives and implicit differentiation. 2. **Partial Derivatives:** These are derivatives of multivariable functions with respect to one variable while keeping other variables constant. For example, if $z = f(x,y)$, then the partial derivative with respect to $x$ is: $$\frac{\partial z}{\partial x} = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}$$ This measures how $z$ changes as $x$ changes, with $y$ fixed. 3. **Implicit Differentiation:** This technique is used when a function is given implicitly rather than explicitly. For example, if $F(x,y)=0$ defines $y$ implicitly as a function of $x$, we differentiate both sides with respect to $x$, treating $y$ as a function of $x$ and applying the chain rule. For instance: $$\frac{d}{dx}F(x,y) = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx} = 0$$ Then solve for $\frac{dy}{dx}$. 4. **Key Difference:** Partial derivatives treat other variables as constants and focus on one variable's influence on the function. Implicit differentiation works with equations defining variables implicitly and finds derivatives by differentiating across all variables considering their interdependence. The concise difference: - Partial derivatives: derivatives with respect to one variable holding others constant. - Implicit differentiation: differentiating equations where variables are intertwined, to find derivatives of dependent variables indirectly.