Implicit differentiation is a powerful technique for finding derivatives of complex functions. It allows us to differentiate equations without isolating variables, making it especially useful for equations that are hard to solve explicitly.

This method builds on our understanding of partial derivatives and the chain rule. By applying these concepts to implicitly defined functions, we can tackle more advanced problems in multivariable calculus and real-world applications.

Implicit Differentiation

Differentiating Implicitly Defined Functions

• Implicit functions define relationships between variables without explicitly solving for one variable in terms of the others
• Implicit differentiation differentiates both sides of an implicit equation with respect to an independent variable
• Applies the chain rule to find the derivative of the dependent variable with respect to the independent variable
• Useful for finding the derivative of functions that are not easily solved for the dependent variable ($y$ in terms of $x$)

Applying the Chain Rule and Total Differential

• The chain rule for partial derivatives states that if $z=f(x,y)$ and $x=g(t)$ and $y=h(t)$, then $\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$
• The total differential of a function $f(x,y)$ is $df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$
• Represents the infinitesimal change in $f$ resulting from infinitesimal changes in $x$ and $y$
• Useful for approximating small changes in a function and for implicit differentiation

Implicit Functions and Curves

Implicit Function Theorem and Level Curves

• The implicit function theorem states that if $F(x,y)=0$ and certain conditions are met, then there exists a unique function $y=f(x)$ such that $F(x,f(x))=0$ in some neighborhood of a point $(a,b)$
• Guarantees the existence of an implicit function locally under certain conditions ($\frac{\partial F}{\partial y} \neq 0$)
• Level curves are curves in the $xy$-plane along which a function $f(x,y)$ is constant
• Defined by the equation $f(x,y)=c$ for some constant $c$
• Examples of level curves include contour lines on a topographic map and iso-pressure curves on a weather map

Gradient Vector and Its Applications

• The gradient vector of a function $f(x,y)$ is $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$
• Points in the direction of the greatest rate of increase of $f$ at a given point
• Perpendicular to the level curve of $f$ at any point
• Useful for finding the direction of steepest ascent or descent (e.g., in optimization problems or in studying the flow of fluids or heat)